# Asymptotic $\psi$-density of subsets of
positive integers

J. Heittokangas and Z. Latreuch

###### Abstract.

The sizes of subsets of the natural numbers are typically quantified in terms
of asymptotic (linear) and logarithmic densities. These concepts have been
generalized to weighted $w$-densities, where a specific weight function $w$
plays a key role. In this paper, a parallel theory of asymptotic
$\psi$-densities is introduced, where the weight is expressed slightly
differently in terms of differentiable functions $\psi$, which are either
concave or convex and satisfy certain asymptotic properties. Alternative new
proofs for known results on analytic and Abel densities are also given.
Keywords: Abel density, analytic density, asymptotic density, asymptotic
$\psi$-density, logarithmic density, regular density, weighted density.
2020 MSC: Primary 11B05, Secondary 26A51.

11footnotetext: Corresponding author: J. Heittokangas.22footnotetext: Z.
Latreuch was supported by the University of Mostaganem and the National Higher
School of Mathematics. He would like to thank the Department of Physics and
Mathematics at the UEF for its hospitality during his visit in March 2023.

##  1\. Introduction

The well-known asymptotic and logarithmic densities of a subset $A$ of the set
$\mathbb{N}$ of natural numbers are given, respectively, by

$\operatorname{d}(A)=\lim_{n\to\infty}\frac{\sum_{k=1}^{n}\chi_{A}(k)}{n}\quad\text{and}\quad\operatorname{\delta}(A)=\lim_{n\to\infty}\frac{\sum_{k=1}^{n}\frac{\chi_{A}(k)}{k}}{\log
n}$

provided that the limits exist. Here $\chi_{A}$ denotes the characteristic
function of the set $A$. Replacing the limits with limit inferiors or with
limit superiors, which always exist, we get the corresponding lower densities
$\underline{\operatorname{d}}(A),\,\underline{\operatorname{\delta}}(A)$ and
the upper densities
$\overline{\operatorname{d}}(A),\,\overline{\operatorname{\delta}}(A)$. These
quantities are related to each other by means of the well-known inequalities

$0\leq\underline{\operatorname{d}}(A)\leq\underline{\operatorname{\delta}}(A)\leq\overline{\operatorname{\delta}}(A)\leq\overline{\operatorname{d}}(A)\leq
1,$ (1.1)

see [7, p. 241] or [18, p. 416]. In the literature, the asymptotic density is
also known by the names of linear density and natural density.

Being able to quantify the sizes of sets $A\subset\mathbb{N}$ is particularly
important in (analytic) number theory [10, 18] and in the theories of
sequences [7] and series [12]. As for a simple example, if $A$ is the set of
all even (or odd) natural numbers and if $B$ is the set of all prime numbers,
then $\operatorname{d}(A)=1/2$ and $\operatorname{d}(B)=0$. The latter is a
simple consequence of the celebrated prime number theorem.

Naturally, the situation calls for a more advanced quantification of sets. The
concept of a weighted density, introduced in [1, 14], provides a
generalization of both asymptotic and logarithmic densities. Indeed, a
weighted density is determined by a function $w:\mathbb{N}\to(0,\infty)$,
known as a weight function, which satisfies the conditions

$\sum_{k=1}^{\infty}w(k)=\infty\quad\text{and}\quad\lim_{n\to\infty}\frac{w(n)}{\sum_{k=1}^{n}w(k)}=0.$
(1.2)

Given a set $A\subset\mathbb{N}$, the lower and upper weighted $w$-densities
with respect to $w$ are defined, respectively, by

$\displaystyle\underline{\operatorname{d}}_{w}(A)=\liminf_{n\to\infty}\frac{\sum_{k=1}^{n}w(k)\chi_{A}(k)}{\sum_{k=1}^{n}w(k)}\quad\text{and}\quad\overline{\operatorname{d}}_{w}(A)=\limsup_{n\to\infty}\frac{\sum_{k=1}^{n}w(k)\chi_{A}(k)}{\sum_{k=1}^{n}w(k)},$
(1.3)

Furthermore, if $f$ and $g$ are two weight functions such that the ratio
$f(n)/g(n)$ is a decreasing function, then the following inequalities hold for
any set $A\subset\mathbb{N}$:

$0\leq\underline{\operatorname{d}}_{g}(A)\leq\underline{\operatorname{d}}_{f}(A)\leq\overline{\operatorname{d}}_{f}(A)\leq\overline{\operatorname{d}}_{g}(A)\leq
1.$ (1.4)

These inequalities can be derived from the following Rajagopal’s classical
result.

######  Theorem 1.1 ([13, Theorem 3]).

Let $(s_{n})_{n}$ be a real sequence, and let $(a_{n})_{n}$ and $(b_{n})_{n}$
be positive real sequences such that

$A_{n}=\sum_{k=1}^{n}a_{k}\to\infty,\quad
B_{n}=\sum_{k=1}^{n}b_{n}\to\infty,\quad n\to\infty.$

Denote

$\sigma(s_{n},a_{n})=\frac{\sum_{k=1}^{n}a_{n}s_{n}}{A_{n}}\quad\text{and}\quad\sigma(s_{n},b_{n})=\frac{\sum_{k=1}^{n}b_{n}s_{n}}{B_{n}}.$

If $(a_{n}/b_{n})_{n}$ is decreasing, then

$\liminf_{n\to\infty}\sigma(s_{n},b_{n})\leq\liminf_{n\to\infty}\sigma(s_{n},a_{n})\leq\limsup_{n\to\infty}\sigma(s_{n},a_{n})\leq\limsup_{n\to\infty}\sigma(s_{n},b_{n}).$

It is worth noting that when considering a real subset $E$ of the interval
$\left(r_{0},R\right)$, where $0<r_{0}<R\leq\infty$, instead of an integer
subset $A\subset\mathbb{N}$, the lower and upper $\psi$-densities of
$E\subset\left(r_{0},R\right)$ are defined [3], respectively, by

$\displaystyle\underline{\psi-\operatorname{dens}}(E)=\liminf_{r\rightarrow
R^{-}}\frac{\int_{E\cap\left[r_{0},r\right)}d\psi(t)}{\psi(r)},\quad\overline{\psi-\operatorname{dens}}(E)=\limsup_{r\rightarrow
R^{-}}\frac{\int_{E\cap\left[r_{0},r\right)}d\psi(t)}{\psi(r)}.$ (1.5)

In this context, $\psi$ is a positive, unbounded, differentiable, and strictly
increasing function on $\left(r_{0},R\right)$. Analogously to (1.4), it has
been shown in [3, Lemma 1] that

$\displaystyle
0\leq\underline{\psi-\operatorname{dens}}(E)\leq\underline{\phi-\operatorname{dens}}(E)\leq\overline{\phi-\operatorname{dens}}(E)\leq\overline{\psi-\operatorname{dens}}(E)\leq
1,$ (1.6)

assuming that $\psi$ is convex with respect to $\varphi$. Note that when the
limit inferiors and superiors coincide in (1.5), the $\psi$-density of $E$ can
be expressed as

$\displaystyle\psi-\operatorname{dens}(E)=\lim_{r\rightarrow
R^{-}}\frac{\int_{E\cap\left[r_{0},r\right)}d\psi(t)}{\psi(r)}=\lim_{r\rightarrow
R^{-}}\frac{\int_{E\cap[r_{0},r)}\psi^{\prime}(t)\,dt}{\int_{[r_{0},r)}\psi^{\prime}(t)\,dt}.$
(1.7)

By considering $E\subset\mathbb{N}$ and $R=\infty$ in (1.7), and substituting
Lebesgue measure with the counting measure, we obtain

$\displaystyle\psi-\operatorname{dens}(E)=\lim_{n\rightarrow\infty}\frac{\sum_{k\in
E\cap\mathbb{N}_{n}}\psi^{\prime}(k)}{\sum_{k=1}^{n}\psi^{\prime}(k)}=\lim_{n\rightarrow\infty}\frac{\sum_{k=1}^{n}\psi^{\prime}(k)\chi_{E}(k)}{\sum_{k=1}^{n}\psi^{\prime}(k)},$
(1.8)

where $\mathbb{N}_{n}=\\{1,\ldots,n\\}$. Assuming further that $\psi$
satisfies the asymptotic identity

$\displaystyle\sum_{k=1}^{n}\psi^{\prime}(k)\sim\psi(n),\quad n\to\infty,$
(1.9)

we then have

$\displaystyle\psi-\operatorname{dens}(E)=\lim_{n\rightarrow\infty}\frac{\sum_{k=1}^{n}\psi^{\prime}(k)\chi_{E}(k)}{\psi(n)},$
(1.10)

which bears a stronger resemblance to (1.5) than to (1.3).

Motivated by (1.8) and (1.10), we propose the following definition for the
weighted density of integer sets that is based on strictly increasing
differentiable functions, typically satisfying (1.9), which takes the role of
the second condition in (1.2).

######  Definition 1.2.

Let $\psi:(0,\infty)\to(0,\infty)$ be a strictly increasing, differentiable,
and unbounded function. The lower and upper $\psi$-densities of a set
$A\subset\mathbb{N}$ are defined, respectively, by

$\displaystyle\underline{\operatorname{d}}_{\psi}(A)=\liminf_{n\to\infty}\frac{\sum_{k=1}^{n}\psi^{\prime}(k)\chi_{A}(k)}{\sum_{k=1}^{n}\psi^{\prime}(k)}\quad\textnormal{and}\quad\overline{\operatorname{d}}_{\psi}(A)=\limsup_{n\to\infty}\frac{\sum_{k=1}^{n}\psi^{\prime}(k)\chi_{A}(k)}{\sum_{k=1}^{n}\psi^{\prime}(k)}.$

If
$\underline{\operatorname{d}}_{\psi}(A)=\overline{\operatorname{d}}_{\psi}(A)$,
then their common value $\operatorname{d}_{\psi}(A)$ is called the
$\psi$-density of $A$. Additionally, if $\psi$ satisfies (1.9), then
$\operatorname{d}_{\psi}(A)$ is called the asymptotic $\psi$-density of $A$.

Based on the fact that $\psi^{\prime}(x)>0$ for all $x>0$, it is evident that
for any set $A\subset\mathbb{N}$, the following inequalities hold

$0\leq\underline{\operatorname{d}}_{\psi}(A)\leq\overline{\operatorname{d}}_{\psi}(A)\leq
1.$

In addition, we have

$\overline{\operatorname{d}}_{\psi}(A^{c})+\underline{\operatorname{d}}_{\psi}(A)=1,$

where $A^{c}$ represents the complement of $A$ in $\mathbb{N}$. Note that both
the $\psi$-density and the asymptotic $\psi$-density satisfy the axiomatic
definition of a density, as proposed in [6]. We point out here that
$\psi$-density and the asymptotic $\psi$-density can be defined without
requiring $\psi$ to be unbounded. For instance, consider the function

$\psi(x)=\frac{1}{e-1}-e^{-x},$

which is a positive, strictly increasing, differentiable, and bounded function
on $[1,\infty)$. It also satisfies (1.9). Such functions may lead to the
corresponding $\psi$-density and asymptotic $\psi$-density not satisfying the
finiteness axiom, which requires that for any finite subset $A$ of
$\mathbb{N}$, we have $\operatorname{d}_{\psi}(A)=0$.

Observe that the asymptotic $\psi$-density aligns with the asymptotic (linear)
and logarithmic densities when $\psi(x)=x$ and $\psi(x)=\log x$, respectively.
In these two cases, the asymptotic $\psi$-density gives a better justification
for the names of the densities than the weighted $w$-density. Also note that
the logarithmic density is in fact an asymptotic $\log$-density, because it is
well known from elementary real analysis that the function $\psi(x)=\log x$
satisfies (1.9).

The rest of this paper is organized as follows. In Section 2 we introduce two
classes of functions, $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$, based on
concavity and convexity, such that the functions $\psi$ in these classes
satisfy (1.9). We then proceed to prove a generalization of (1.1) for
asymptotic $\psi$-densities, and also discuss the equivalence in existence of
two given asymptotic $\psi$-densities. In Section 3 we discuss the asymptotic
$\psi$-density of strictly increasing sequences. In Section 4 we consider the
regularity of asymptotic $\psi$-densities for functions $\psi$ in the classes
$\mathcal{D}_{1}$ and $\mathcal{D}_{2}$. A function $\psi$ and the
corresponding asymptotic $\psi$-density $\operatorname{d}_{\psi}$ are called
regular, if $\operatorname{d}_{\psi}(\mathcal{A})=1/a$ for any integer set
$\mathcal{A}$ that consists of arithmetic progressions of the form $an+b$,
where $a$ and $b$ are fixed positive integers. This paper is concluded by two
appendices that contain clarifying discussions as well as alternative proofs
of known results involving analytic and Abel densities.

##  2\. Asymptotic $\psi$-densities

We introduce a function class $\mathcal{D}_{1}$ of concave functions with
certain properties, and a function class $\mathcal{D}_{2}$ of convex functions
with corresponding properties. The functions $\psi$ in these two classes
satisfy (1.9), which defines the concept of asymptotic $\psi$-density. We then
proceed to give a comparison between asymptotic densities for two distinct
weight functions $\psi$ and $\varphi$, and study the equivalence of these two
densities.

###  2.1. Function classes $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$

We begin by defining a class $\mathcal{D}$ of strictly increasing,
differentiable, and unbounded functions $\psi:(0,\infty)\to(0,\infty)$. For
convenience, we set

$A_{\psi}(n):=\sum_{k=1}^{n}\psi^{\prime}(k)\chi_{A}(k)$ (2.1)

for any subset $A\subset\mathbb{N}$. Then, if $\psi\in\mathcal{D}$ satisfies
(1.9), the lower and upper asymptotic $\psi$-densities can be expressed more
briefly as

$\underline{\operatorname{d}}_{\psi}(A)=\liminf_{n\to\infty}\frac{A_{\psi}(n)}{\psi(n)}\quad\text{and}\quad\overline{\operatorname{d}}_{\psi}(A)=\limsup_{n\to\infty}\frac{A_{\psi}(n)}{\psi(n)}.$

At times, we consider the above $A_{\psi}$ as a step function, defined for the
real variable $x$ by

$A_{\psi}(x)=\begin{cases}\sum_{1\leq k\leq
x}\psi^{\prime}(k)\chi_{A}(k),&\text{if }x\geq 1,\\\ 0,&\text{if }0\leq
x<1.\end{cases}$

We proceed to discuss which strictly increasing, differentiable, and unbounded
functions $\psi$ satisfy (1.9). To this end, we introduce two classes of
functions based on concavity and convexity.

######  Definition 2.1.

Let $\mathcal{D}_{1}$ denote the class of all concave functions in
$\mathcal{D}$, and let $\mathcal{D}_{2}$ denote the class of all convex
functions $\psi$ in $\mathcal{D}$ that satisfy

$\psi(n+1)\sim\psi(n),\quad n\to\infty.$ (2.2)

The classes $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$ are certainly not empty:
The function $\psi_{1}(x)=\log(1+x)$ belongs to $\mathcal{D}_{1}$, while
$\psi_{2}(x)=x^{2}$ belongs to $\mathcal{D}_{2}$.

The reason for the interval $(0,\infty)$ to be the domain of definition for
functions in $\mathcal{D}$ is nothing but technical. It would be acceptable to
replace this interval by $(r_{0},\infty)$, where $r_{0}>0$ is some constant.
This would only cause some minor modifications in the reasoning below. Hence,
the function $\psi(x)=\log x$, for example, would be acceptable for the class
$\mathcal{D}_{1}$. The assumption $\psi(x)>0$ cannot be ignored, however.

Lemma 2.2 below shows that all functions $\psi\in\mathcal{D}_{1}$ satisfy
(2.2). However, one should note that not all convex functions satisfy (2.2),
as illustrated by the counter-example $\psi(x)=e^{x}$.

######  Lemma 2.2.

If $\psi\in\mathcal{D}_{1}$, then (1.9) and (2.2) both hold. If
$\psi\in\mathcal{D}_{2}$, then (1.9) holds.

###### Proof.

Suppose that $\psi\in\mathcal{D}_{1}$. Then $\psi^{\prime}$ is a decreasing
function, and, by the endpoint rules of integral calculus,

$\displaystyle\psi(n+1)-\psi(1)$ $\displaystyle=$
$\displaystyle\int_{1}^{n+1}\psi^{\prime}(x)\,dx\leq\sum_{k=1}^{n}\psi^{\prime}(k)$
$\displaystyle\leq$
$\displaystyle\psi^{\prime}(1)+\int_{1}^{n}\psi^{\prime}(x)\,dx=\psi(n)+\psi^{\prime}(1)-\psi(1).$

Since the upper bound is $\leq\psi(n+1)+\psi^{\prime}(1)-\psi(1)$, we obtain
(1.9) and (2.2).

Suppose then that $\psi\in\mathcal{D}_{2}$. Then $\psi^{\prime}$ is an
increasing function, and

$\displaystyle\psi(n)+\psi^{\prime}(1)-\psi(1)$ $\displaystyle=$
$\displaystyle\psi^{\prime}(1)+\int_{1}^{n}\psi^{\prime}(x)\,dx\leq\sum_{k=1}^{n}\psi^{\prime}(k)$
$\displaystyle\leq$
$\displaystyle\int_{1}^{n+1}\psi^{\prime}(x)\,dx=\psi(n+1)-\psi(1).$

The assertion (1.9) follows from this via the assumption (2.2). ∎

Remark. Lemma 2.2 ensures that $\operatorname{d}_{\psi}$ is an asymptotic
$\psi$-density for functions $\psi\in\mathcal{D}_{1}\cup\mathcal{D}_{2}$. In
fact, this property justifies the definition of the classes $\mathcal{D}_{1}$
and $\mathcal{D}_{2}$.

######  Example 2.3.

Using Lemma 2.2, we can obtain the known asymptotic growth of $p$-sums.
Indeed, let $\psi(x)=x^{p+1}$ for $x>0$ and $p>-1$. Then
$\psi\in\mathcal{D}_{1}$ for $-1<p\leq 0$ and $\psi\in\mathcal{D}_{2}$ for
$p\geq 0$. Moreover, by Lemma 2.2,

$\sum_{k=1}^{n}k^{p}=\frac{1}{p+1}\sum_{k=1}^{n}\psi^{\prime}(k)\sim\frac{\psi(n)}{p+1}=\frac{n^{p+1}}{p+1},\quad
p>-1.$

For $\psi\in\mathcal{D}_{1}$, we can replace $n$ by a real variable $x$ in the
proof of Lemma 2.2, and obtain

$\psi(x+1)\sim\psi(x),\quad x\to\infty.$ (2.3)

To prove that (2.3) holds for functions $\psi\in\mathcal{D}_{2}$ also, we
first require a lemma that will be needed several times later on.

######  Lemma 2.4.

Let $\psi:(0,\infty)\to(0,\infty)$ be an increasing, differentiable and convex
function. Then

  * (a)

(2.2) holds if and only if $\psi^{\prime}(n)/\psi(n)\to 0$ as $n\to\infty$,

  * (b)

(2.3) holds if and only if $\psi^{\prime}(x)/\psi(x)\to 0$ as $x\to\infty$.

###### Proof.

It suffices to prove (b) because the proof of (a) is the same.

Assume that $\psi(x+1)\sim\psi(x)$. By utilizing the convexity of $\psi$ and
the mean value theorem, we have

$\displaystyle\psi^{\prime}(x)\leq\psi(x+1)-\psi(x)\leq\psi^{\prime}(x+1),$
(2.4)

for any $x\in(0,\infty)$. Dividing both sides of (2.4) by $\psi(x)$, we obtain

$0\leq\frac{\psi^{\prime}(x)}{\psi(x)}\leq\frac{\psi(x+1)}{\psi(x)}-1,$

and the assertion $\psi^{\prime}(x)/\psi(x)\to 0$ as $x\to\infty$ follows by
the squeeze theorem.

Conversely, assume that $\psi^{\prime}(x)/\psi(x)\to 0$ as $x\to\infty$.
Dividing both sides of (2.4) by $\psi(x+1)$ yields

$\frac{\psi^{\prime}(x+1)}{\psi(x+1)}\geq 1-\frac{\psi(x)}{\psi(x+1)}\geq 0.$

By letting again $x\to\infty$, we derive (2.3). ∎

######  Lemma 2.5.

If $\psi\in\mathcal{D}_{2}$, then (2.3) holds.

###### Proof.

Suppose on the contrary to the assertion that (2.3) does not hold. Then, by
Lemma 2.4, the logarithmic derivative $\psi^{\prime}(x)/\psi(x)$ does not tend
to zero along the real numbers, even though $\psi^{\prime}(n)/\psi(n)$ does
tend to zero along the natural numbers. Therefore, there is a strictly
increasing and unbounded sequence $(x_{n})_{n}$ of positive real numbers and a
constant $\delta>0$ such that

$\psi^{\prime}(x_{n})/\psi(x_{n})>\delta,\quad n\in\mathbb{N}.$

Clearly, the sequence $(x_{n})_{n}$ has at most a finite number of points
$x_{n}$ that are natural numbers. We may assume that these points, if any,
have been discarded from the sequence $(x_{n})_{n}$. Now, for every $x_{n}$
there is a unique $k_{n}\in\mathbb{N}$ such that $x_{n}\in(k_{n},k_{n}+1)$.
Then, by the monotonicity and convexity of $\psi$,

$0<\delta<\frac{\psi^{\prime}(x_{n})}{\psi(x_{n})}\leq\frac{\psi^{\prime}(k_{n}+1)}{\psi(k_{n})}\to
0,\quad n\to\infty,$

which is a contradiction. Thus (2.3) holds. ∎

A further discussion on functions $\psi$ satisfying (2.3) will be given in
Section 4.2 and in Appendix A below. We show, among other things, that there
are strictly increasing, differentiable and convex functions $\psi$ that do
not satisfy (2.3), and also ponder the strength of the asymptotic identity
(2.3) for certain functions $\psi$. This discussion can be of independent
interest.

###  2.2. Comparison of asymptotic $\psi$-densities

We proceed with recalling the concepts of relative convexity and concavity.
Let $\psi$ and $\varphi$ be functions in $\mathcal{D}$. We say that $\varphi$
is convex with respect to $\psi$ if the function $\varphi\circ\psi^{-1}$ is
convex. Equivalently, $\varphi$ is convex with respect to $\psi$ (or $\psi$ is
concave with respect to $\varphi$) if the quotient
$\varphi^{\prime}(x)/\psi^{\prime}(x)$ is increasing for all $x\in(0,\infty)$,
as shown in [4, Theorem 1]. In other words, if both $\psi$ and $\varphi$ are
twice differentiable functions, then it follows that

$\left(\frac{\psi^{\prime}(x)}{\varphi^{\prime}(x)}\right)^{\prime}\leq
0,\quad x\in(0,\infty).$

We are now ready to present a result which is similar to (1.4) and (1.6).

######  Theorem 2.6.

Let $A\subset\mathbb{N}$, and suppose that $\psi,\varphi\in\mathcal{D}$
satisfy (1.9). If $\psi$ is concave with respect to $\varphi$, then

$\displaystyle
0\leq\underline{\operatorname{d}}_{\varphi}(A)\leq\underline{\operatorname{d}}_{\psi}(A)\leq\overline{\operatorname{d}}_{\psi}(A)\leq\overline{\operatorname{d}}_{\varphi}(A)\leq
1.$ (2.5)

As per the terminology introduced in the literature, the two densities
satisfying (2.5) are comparable. More specifically, the $\varphi$-density is
stronger than the $\psi$-density, and the $\psi$-density extends the
$\varphi$-density.

For the convenience of the reader, we present two distinct proofs for Theorem
2.6. The first proof utilizes Rajagopal’s result, while the second one relies
on Riemann-Stieltjes integration, but requires the assumption that $\psi$ and
$\varphi$ are both twice differentiable. The technique used in the second
proof will be applied later on in this paper.

###### First proof of Theorem 2.6.

Given that $\psi$ and $\varphi$ belong to $\mathcal{D}$ and satisfy (1.9), we
observe that

$\sum_{k=1}^{n}\psi^{\prime}(k)\to\infty\quad\text{and}\quad\sum_{k=1}^{n}\varphi^{\prime}(k)\to\infty,\quad
n\to\infty.$

On the other hand, due to the fact that $\psi$ is concave with respect to
$\varphi$, we conclude that $\psi^{\prime}(x)/\varphi^{\prime}(x)$ is
decreasing. Therefore, by using Theorem 1.1, we establish the assertion (2.5).
∎

###### Second proof of Theorem 2.6.

We make an additional assumption that $\psi$ and $\varphi$ are both twice
differentiable. It suffices to prove that

$\underline{\operatorname{d}}_{\varphi}(A)\leq\underline{\operatorname{d}}_{\psi}(A)\quad\textnormal{and}\quad\overline{\operatorname{d}}_{\psi}(A)\leq\overline{\operatorname{d}}_{\varphi}(A).$

The first inequality is trivial if
$\underline{\operatorname{d}}_{\varphi}(A)=0$, so we suppose that
$\underline{\operatorname{d}}_{\varphi}(A)>0$. The proof of the second
inequality above does not depend on this restriction.

Applying Riemann-Stieltjes integration to the step function $A_{\varphi}(x)$
yields

$\displaystyle A_{\psi}(n)$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{n}\psi^{\prime}(k)\chi_{A}(k)=\sum_{k=1}^{n}\frac{\psi^{\prime}(k)}{\varphi^{\prime}(k)}\varphi^{\prime}(k)\chi_{A}(k)$
$\displaystyle=$
$\displaystyle\sum_{k=1}^{n}\frac{\psi^{\prime}(k)}{\varphi^{\prime}(k)}\big{(}A_{\varphi}(k)-A_{\varphi}(k-1)\big{)}=\int_{1}^{n}\frac{\psi^{\prime}(t)}{\varphi^{\prime}(t)}dA_{\varphi}(t),$

see [2, Theorem 7.9] and [2, Theorem 7.11]. Integrating by parts, we get

$\displaystyle\begin{split}A_{\psi}(n)&=\frac{\psi^{\prime}(n)}{\varphi^{\prime}(n)}A_{\varphi}(n)-\int_{1}^{n}\left(\frac{\psi^{\prime}(t)}{\varphi^{\prime}(t)}\right)^{\prime}A_{\varphi}(t)\,dt+O(1).\end{split}$
(2.6)

Let $\varepsilon\in(0,\underline{\operatorname{d}}_{\varphi}(A))$. Then there
exists a $t_{0}=t_{0}(\varepsilon)>0$ such that

$\underline{\operatorname{d}}_{\varphi}(A)-\varepsilon\leq\frac{A_{\varphi}(t)}{\varphi(t)}\leq\overline{\operatorname{d}}_{\varphi}(A)+\varepsilon,\quad
t\geq t_{0}.$

Making use of this, we obtain, for all large $n\geq t_{0}$,

$\displaystyle\left(\underline{\operatorname{d}}_{\varphi}(A)-\varepsilon\right)\frac{\psi^{\prime}(n)}{\varphi^{\prime}(n)}\varphi(n)\leq\frac{\psi^{\prime}(n)}{\varphi^{\prime}(n)}A_{\varphi}(n)\leq\left(\overline{\operatorname{d}}_{\varphi}(A)+\varepsilon\right)\frac{\psi^{\prime}(n)}{\varphi^{\prime}(n)}\varphi(n)$
(2.7)

and

$\begin{split}\left(\underline{\operatorname{d}}_{\varphi}(A)-\varepsilon\right)&\int_{1}^{n}\left(-\frac{\psi^{\prime}(t)}{\varphi^{\prime}(t)}\right)^{\prime}\varphi(t)dt+O(1)\leq\int_{1}^{n}\left(-\frac{\psi^{\prime}(t)}{\varphi^{\prime}(t)}\right)^{\prime}A_{\varphi}(t)dt\\\
&\leq\left(\overline{\operatorname{d}}_{\varphi}(A)+\varepsilon\right)\int_{1}^{n}\left(-\frac{\psi^{\prime}(t)}{\varphi^{\prime}(t)}\right)^{\prime}\varphi(t)dt+O(1),\end{split}$
(2.8)

where
$\left(-\frac{\psi^{\prime}(t)}{\varphi^{\prime}(t)}\right)^{\prime}\varphi(t)\geq
0$ for all $t>0$. A further integration by parts gives us

$-\int_{1}^{n}\left(\frac{\psi^{\prime}(t)}{\varphi^{\prime}(t)}\right)^{\prime}\varphi(t)dt=\psi(n)\left(1-\frac{\psi^{\prime}(n)}{\psi(n)}\frac{\varphi(n)}{\varphi^{\prime}(n)}\right)+O(1).$

Substituting this into (2.8) leads to

$\begin{split}\left(\underline{\operatorname{d}}_{\varphi}(A)-\varepsilon\right)&\psi(n)\left(1-\frac{\psi^{\prime}(n)}{\psi(n)}\frac{\varphi(n)}{\varphi^{\prime}(n)}\right)+O(1)\leq\int_{1}^{n}\left(-\frac{\psi^{\prime}(t)}{\varphi^{\prime}(t)}\right)^{\prime}A_{\varphi}(t)dt\\\
&\leq\left(\overline{\operatorname{d}}_{\varphi}(A)+\varepsilon\right)\psi(n)\left(1-\frac{\psi^{\prime}(n)}{\psi(n)}\frac{\varphi(n)}{\varphi^{\prime}(n)}\right)+O(1),\quad
t\geq t_{0}.\end{split}$

Combining this with (2.6) and (2.7), we find after simple calculations that
for all large $n\geq t_{0}$,

$\left(\underline{\operatorname{d}}_{\varphi}(A)-\varepsilon\right)+o(1)\leq\frac{A_{\psi}(n)}{\psi(n)}\leq\left(\overline{\operatorname{d}}_{\varphi}(A)+\varepsilon\right)+o(1).$

Taking limit inferior on the first inequality and limit superior on the second
inequality gives

$\underline{\operatorname{d}}_{\varphi}(A)-\varepsilon\leq\underline{\operatorname{d}}_{\psi}(A)\quad\textnormal{and}\quad\overline{\operatorname{d}}_{\psi}(A)\leq\overline{\operatorname{d}}_{\varphi}(A)+\varepsilon.$

The assertions follow by letting $\varepsilon\to 0^{+}$. ∎

Since any concave function $\psi$ is concave with respect to the identity
function, it follows that the asymptotic density can be thought of as the
“terminal density” for the asymptotic $\psi$-densities when
$\psi\in\mathcal{D}_{1}$. Likewise, any convex function $\varphi$ is convex
with respect to the identity function. Therefore, the asymptotic density is
the “initial density” for the asymptotic $\varphi$-densities when
$\varphi\in\mathcal{D}_{2}$. These observations will be made more precise in
the following immediate consequence of Theorem 2.6.

######  Corollary 2.7.

The following assertions hold for any $A\subset\mathbb{N}$.

  * (a)

If $\psi\in\mathcal{D}_{1}$, then

$0\leq\underline{\operatorname{d}}(A)\leq\underline{\operatorname{d}}_{\psi}(A)\leq\overline{\operatorname{d}}_{\psi}(A)\leq\overline{\operatorname{d}}(A)\leq
1.$

  * (b)

If $\varphi\in\mathcal{D}_{2}$, then

$0\leq\underline{\operatorname{d}}_{\varphi}(A)\leq\underline{\operatorname{d}}(A)\leq\overline{\operatorname{d}}(A)\leq\overline{\operatorname{d}}_{\varphi}(A)\leq
1.$

###  2.3. Comparison of other densities

For $p>0$, set

$\zeta(p):=\sum_{n=1}^{\infty}\frac{1}{n^{p}}.$

It is well-known that $\zeta(p)<\infty$ for all $p>1$ and that
$\zeta(p)=\infty$ for all $0<p\leq 1$. The analytic density of a set
$A\subset\mathbb{N}$ is defined in [18, p. 418] as

$\displaystyle\operatorname{d}_{a}(A):=\lim_{p\to
1^{+}}\frac{1}{\zeta(p)}\sum_{n=1}^{\infty}\frac{\chi_{A}(n)}{n^{p}},$

provided that the limit exists. Alternatively (see Appendix B), we may write

$\operatorname{d}_{a}(A)=\lim_{p\to
1^{+}}(p-1)\sum_{n=1}^{\infty}\frac{\chi_{A}(n)}{n^{p}}.$

The analytic density is closely related to the logarithmic density in the
following sense.

######  Theorem 2.8 ([18, p. 418]).

Let $A\subset\mathbb{N}$. Then the analytic density of $A$ exists if and only
if the logarithmic density of $A$ exists. In this case, the two densities are
equal.

We present an alternative proof for Theorem 2.8 in Appendix B. More generally,
it is natural to ask the following question, which will be answered later in
this section.

######  Question 1.

Under what conditions on $\psi,\varphi\in\mathcal{D}$, the $\psi$-density of
$A$ exists if and only if the $\varphi$-density of $A$ exists? If the two
densities exist, are they equal?

The above equivalence fails to hold between asymptotic and logarithmic
densities. Indeed, it is known [18, p. 417] that the existence of the
logarithmic density does not imply the existence of the asymptotic density.
Conversely, the existence of the asymptotic density does imply the existence
of the logarithmic density by (1.1).

The next result gives a comparison between different asymptotic
$\psi$-densities and contains [9, Proposition 1] as a special case.

######  Theorem 2.9.

Let $A\subset\mathbb{N}$, and let $\psi$ and $\varphi$ be functions in
$\mathcal{D}$ that satisfy (1.9). If

$\displaystyle
0<c_{1}=\liminf_{x\rightarrow\infty}\frac{\psi^{\prime}(x)}{\varphi^{\prime}(x)}\leq\limsup_{x\rightarrow\infty}\frac{\psi^{\prime}(x)}{\varphi^{\prime}(x)}=c_{2}<\infty,$
(2.9)

then

$\displaystyle\frac{1}{c}\cdot\underline{\operatorname{d}}_{\varphi}(A)\leq\underline{\operatorname{d}}_{\psi}(A)\leq
c\cdot\underline{\operatorname{d}}_{\varphi}(A)\quad\text{and}\quad\frac{1}{c}\cdot\overline{\operatorname{d}}_{\varphi}(A)\leq\overline{\operatorname{d}}_{\psi}(A)\leq
c\cdot\overline{\operatorname{d}}_{\varphi}(A),$ (2.10)

where $c=c_{2}/c_{1}$. In particular, if $c=1$, then

$\underline{\operatorname{d}}_{\psi}(A)=\underline{\operatorname{d}}_{\varphi}(A)\quad\text{and}\quad\overline{\operatorname{d}}_{\psi}(A)=\overline{\operatorname{d}}_{\varphi}(A).$

###### Proof.

Using (2.9) and [17, Theorem II], we obtain

$0<c_{1}=\liminf_{x\rightarrow\infty}\frac{\psi^{\prime}(x)}{\varphi^{\prime}(x)}\leq\liminf_{x\rightarrow\infty}\frac{\psi(x)}{\varphi(x)}\leq\limsup_{x\rightarrow\infty}\frac{\psi(x)}{\varphi(x)}\leq\limsup_{x\rightarrow\infty}\frac{\psi^{\prime}(x)}{\varphi^{\prime}(x)}=c_{2}<\infty.$

Let $\varepsilon\in(0,c_{1})$. Then there exists an $N=N(\varepsilon)>0$ such
that

$\displaystyle c_{1}-\varepsilon\leq\frac{\psi(x)}{\varphi(x)}\leq
c_{2}+\varepsilon\quad\text{and}\quad
c_{1}-\varepsilon\leq\frac{\psi^{\prime}(x)}{\varphi^{\prime}(x)}\leq
c_{2}+\varepsilon,\quad x\geq N.$ (2.11)

Hence,

$\frac{1}{c_{2}+\varepsilon}\leq\frac{\varphi(x)}{\psi(x)}\leq\frac{1}{c_{1}-\varepsilon}\quad\text{and}\quad\frac{1}{c_{2}+\varepsilon}\leq\frac{\varphi^{\prime}(x)}{\psi^{\prime}(x)}\leq\frac{1}{c_{1}-\varepsilon},\quad
x\geq N.$

Now, let $A\subset\mathbb{N}$ be a set. Then

$\displaystyle A_{\varphi}(n)$ $\displaystyle=$
$\displaystyle\sum_{k=N}^{n}\varphi^{\prime}(k)\chi_{A}(k)+O(1)\leq\frac{1}{c_{1}-\varepsilon}\sum_{k=N}^{n}\psi^{\prime}(k)\chi_{A}(k)+O(1)$
$\displaystyle=$
$\displaystyle\frac{1}{c_{1}-\varepsilon}A_{\psi}(n)+O(1),\quad n\geq N,$

and similarly,

$\displaystyle
A_{\varphi}(n)\geq\frac{1}{c_{2}+\varepsilon}A_{\psi}(n)+O(1),\quad n\geq N.$

Combining these estimates with (2.11), we obtain

$\left(\frac{c_{1}-\varepsilon}{c_{2}+\varepsilon}\right)\frac{A_{\psi}(n)}{\psi(n)}+o(1)\leq\frac{A_{\varphi}(n)}{\varphi(n)}\leq\left(\frac{c_{2}+\varepsilon}{c_{1}-\varepsilon}\right)\frac{A_{\psi}(n)}{\psi(n)}+o(1),\quad
n\geq N.$

By using the properties of limit inferior and limit superior, we get

$\left(\frac{c_{1}-\varepsilon}{c_{2}+\varepsilon}\right)\underline{\operatorname{d}}_{\psi}(A)\leq\underline{\operatorname{d}}_{\varphi}(A)\leq\overline{\operatorname{d}}_{\varphi}(A)\leq\left(\frac{c_{2}+\varepsilon}{c_{1}-\varepsilon}\right)\overline{\operatorname{d}}_{\psi}(A),$

and by letting $\varepsilon\to 0^{+}$, we obtain

$\displaystyle\left(\frac{c_{1}}{c_{2}}\right)\underline{\operatorname{d}}_{\psi}(A)\leq\underline{\operatorname{d}}_{\varphi}(A)\leq\overline{\operatorname{d}}_{\varphi}(A)\leq\left(\frac{c_{2}}{c_{1}}\right)\overline{\operatorname{d}}_{\psi}(A).$
(2.12)

Now, by making use of (2.11) again, one may see that

$\displaystyle
0<\frac{1}{c_{2}}\leq\liminf_{x\rightarrow\infty}\frac{\varphi^{\prime}(x)}{\psi^{\prime}(x)}\leq\limsup_{x\rightarrow\infty}\frac{\varphi^{\prime}(x)}{\psi^{\prime}(x)}\leq\frac{1}{c_{1}}<\infty.$

Hence, by using the same argument as above, we obtain

$\displaystyle\left(\frac{c_{1}}{c_{2}}\right)\underline{\operatorname{d}}_{\varphi}(A)\leq\underline{\operatorname{d}}_{\psi}(A)\leq\overline{\operatorname{d}}_{\psi}(A)\leq\left(\frac{c_{2}}{c_{1}}\right)\overline{\operatorname{d}}_{\varphi}(A).$
(2.13)

Combining (2.12) and (2.13), we conclude (2.10). ∎

The following corollary follows directly from Theorem 2.9, and it gives an
answer to Question 1.

######  Corollary 2.10.

Assume that $A\subset\mathbb{N}$, and let $\psi,\varphi\in\mathcal{D}$ be such
that

$\lim_{x\to\infty}\frac{\psi^{\prime}(x)}{\varphi^{\prime}(x)}=C\in(0,\infty).$
(2.14)

Then $\operatorname{d}_{\psi}(A)$ exists if and only if
$\operatorname{d}_{\varphi}(A)$ exists. In this case, the two densities are
equal.

As mentioned above, the existence of the logarithmic density does not imply
the existence of the asymptotic density. Clearly, the assumption (2.14) fails
to hold if $\psi(x)=x$ and $\varphi(x)=\log x$ (or the other way around).

##  3\. Asymptotic $\psi$-density of strictly increasing sequences

Let $(v_{n})_{n}$ be a strictly increasing sequence of positive integers, and
let $\mathcal{S}=\\{v_{n}:n\in\mathbb{N}\\}$ be the associated set. It is
known [10, 11] that the lower and upper asymptotic density of $\mathcal{S}$
can be expressed as

$\displaystyle\underline{\operatorname{d}}(\mathcal{S})=\liminf_{n\to\infty}\frac{n}{v_{n}}\quad\text{and}\quad\overline{\operatorname{d}}(\mathcal{S})=\limsup_{n\to\infty}\frac{n}{v_{n}}.$
(3.1)

Motivated by this, we ask the following question.

######  Question 2.

What is the asymptotic $\psi$-densities analogue of (3.1)?

The following result addresses this question when
$\psi\in\mathcal{D}_{1}\cup\mathcal{D}_{2}$, and is inspired by the proofs of
[10, Theorem 11.1] and [11, Theorem 2.3].

######  Theorem 3.1.

Let $(v_{n})_{n}$ be a strictly increasing infinite sequence of positive
integers. Suppose that one of the following conditions holds:

  * (a)

$\psi\in\mathcal{D}_{1}$,

  * (b)

$\psi\in\mathcal{D}_{2}$ and $\psi^{\prime}(v_{n})/\psi(n)\to 0$, as
$n\to\infty$.

Then

$\displaystyle\underline{\operatorname{d}}_{\psi}(\mathcal{S})=\liminf_{n\to\infty}\frac{\mathcal{S}_{\psi}(v_{n})}{\psi(v_{n})}\quad\text{and}\quad\overline{\operatorname{d}}_{\psi}(\mathcal{S})=\limsup_{n\to\infty}\frac{\mathcal{S}_{\psi}(v_{n})}{\psi(v_{n})}.$
(3.2)

Before we proceed with the proof, we need to establish an auxiliary result.

######  Lemma 3.2.

Let $(v_{n})_{n}$ be a strictly increasing infinite sequence of positive
integers. If $v_{n}\neq n$ for some $n$, then there exists an $N\in\mathbb{N}$
such that $v_{n}>n$ for all $n\geq N$.

###### Proof.

Suppose on the contrary to the assertion that there exists a strictly
increasing infinite sequence of positive integers $(n_{k})_{k}$ such that
$v_{n_{k}}\leq n_{k}$. Note that $v_{n}\geq n$ for all $n$, as $(v_{n})_{n}$
is a strictly increasing infinite sequence of positive integers. Therefore,
$v_{n_{k}}=n_{k},$ for all integers $k$. Making use of this, we get

$n_{k}-1\leq v_{n_{k}-1}<n_{k},$

which implies that $v_{n_{k}-1}=n_{k}-1$. Inductively, we conclude that for
any $n_{k}$, we have $v_{j}=j$ for all $j\leq n_{k}$. Since $(n_{k})_{k}$ is
strictly increasing, we find that $v_{n}=n$ for all $n$, which is a
contradiction. ∎

###### Proof of Theorem 3.1.

Without loss of generality, we assume that $v_{n}\neq n$. Clearly, since
$(v_{n})_{n}$ is a strictly increasing sequence of positive integers,

$\left(\frac{\mathcal{S}_{\psi}(v_{n})}{\psi(v_{n})}\right)_{n}\,\text{is a
sub-sequence of }\;\left(\frac{\mathcal{S}_{\psi}(n)}{\psi(n)}\right)_{n}.$

Hence

$\liminf_{n\to\infty}\frac{\mathcal{S}_{\psi}(v_{n})}{\psi(v_{n})}\geq\liminf_{n\to\infty}\frac{\mathcal{S}_{\psi}(n)}{\psi(n)}\quad\text{and}\quad\limsup_{n\to\infty}\frac{\mathcal{S}_{\psi}(v_{n})}{\psi(v_{n})}\leq\limsup_{n\to\infty}\frac{\mathcal{S}_{\psi}(n)}{\psi(n)}.$

Regarding the first identity in (3.2), it remains to prove that

$\liminf_{n\to\infty}\frac{\mathcal{S}_{\psi}(v_{n})}{\psi(v_{n})}\leq\liminf_{n\to\infty}\frac{\mathcal{S}_{\psi}(n)}{\psi(n)}.$

To this end, let $(m(n))_{n}$ be a sequence of positive integers such that
$v_{m(n)}$ is the smallest integer in $\mathcal{S}$ that exceeds $n$, that is,
$v_{m(n)-1}\leq n<v_{m(n)}$. Note that $(m(n))_{n}$ is an increasing sequence
(not necessarily strictly increasing) and tends to $\infty$ as $n\to\infty$.
Then

$\mathcal{S}_{\psi}(n)=\sum_{k=1}^{n}\psi^{\prime}(k)\chi_{\mathcal{S}}(k)=\sum_{k=1}^{m(n)-1}\psi^{\prime}(v_{k})$

and

$\displaystyle\frac{\mathcal{S}_{\psi}(v_{m(n)})}{\psi(v_{m(n)})}-\frac{\mathcal{S}_{\psi}(n)}{\psi(n)}=$
$\displaystyle\frac{\mathcal{S}_{\psi}(v_{m(n)})}{\psi(v_{m(n)})}-\frac{\sum_{k=1}^{m(n)-1}\psi^{\prime}(v_{k})}{\psi(n)}\leq\frac{\mathcal{S}_{\psi}(v_{m(n)})}{\psi(n)}-\frac{\sum_{k=1}^{m(n)-1}\psi^{\prime}(v_{k})}{\psi(n)}$
$\displaystyle=$
$\displaystyle\frac{\sum_{k=1}^{m(n)}\psi^{\prime}(v_{k})-\sum_{k=1}^{m(n)-1}\psi^{\prime}(v_{k})}{\psi(n)}=\frac{\psi^{\prime}(v_{m(n)})}{\psi(n)}.$

It follows that

$\displaystyle\frac{\mathcal{S}_{\psi}(v_{m(n)})}{\psi(v_{m(n)})}\leq\frac{\mathcal{S}_{\psi}(n)}{\psi(n)}+\frac{\psi^{\prime}(v_{m(n)})}{\psi(n)}.$
(3.3)

Let $(k(n))_{n}$ be the strictly increasing sub-sequence of $(m(n))_{n}$
consisting of all distinct elements in the sequence $(m(n))_{n}$. We
distinguish two cases.

  * (i)

If $\psi\in\mathcal{D}_{1}$, then the derivative $\psi^{\prime}$ is non-
increasing, and hence $\psi^{\prime}$ is bounded from above. This, together
with (3.3), implies

$\liminf_{n\to\infty}\frac{\mathcal{S}_{\psi}(v_{n})}{\psi(v_{n})}\leq\liminf_{n\to\infty}\frac{\mathcal{S}_{\psi}(v_{k(n)})}{\psi(v_{k(n)})}=\liminf_{n\to\infty}\frac{\mathcal{S}_{\psi}(v_{m(n)})}{\psi(v_{m(n)})}\leq\liminf_{n\to\infty}\frac{\mathcal{S}_{\psi}(n)}{\psi(n)},$

and the inequality

$\displaystyle\underline{\operatorname{d}}_{\psi}(\mathcal{S})\geq\liminf_{n\to\infty}\frac{\mathcal{S}_{\psi}(v_{n})}{\psi(v_{n})}$
(3.4)

follows immediately.

  * (ii)

If $\psi\in\mathcal{D}_{2}$, then, by using Lemma 3.2 and considering that
$v_{m(n)}$ is the smallest integer that exceeds $n$, we deduce the existence
of $N\in\mathbb{N}$ such that

$\psi^{\prime}(v_{m(n)})\leq\psi^{\prime}(v_{n}),\quad\text{for all}\;n\geq
N.$

This, along with the assumption $\psi^{\prime}(v_{n})/\psi(n)\to 0$ as
$n\to\infty$, yields

$\lim_{n\to\infty}\frac{\psi^{\prime}(v_{m(n)})}{\psi(n)}=0.$

Consequently, by applying a similar proof to that in Case (i), we obtain
(3.4).

Regarding the second identity in (3.2), it remains to prove that

$\limsup_{n\to\infty}\frac{\mathcal{S}_{\psi}(v_{n})}{\psi(v_{n})}\geq\limsup_{n\to\infty}\frac{\mathcal{S}_{\psi}(n)}{\psi(n)}=\overline{\operatorname{d}}_{\psi}(\mathcal{S}).$

Clearly, we may suppose that
$\overline{\operatorname{d}}_{\psi}(\mathcal{S})>0$. By definition, for any
given $\varepsilon\in(0,\overline{\operatorname{d}}_{\psi}(\mathcal{S}))$,
there exists a strictly increasing sequence $(n_{k})_{k}$ of positive integers
such that

$\displaystyle\frac{\mathcal{S}_{\psi}(n_{k})}{\psi(n_{k})}>\overline{\operatorname{d}}_{\psi}(\mathcal{S})-\varepsilon.$
(3.5)

Let $n_{k_{1}}\geq v_{1}$ be an integer satisfying (3.5). Then there exists a
positive integer $m_{1}$ such that $v_{m_{1}}\leq n_{k_{1}}<v_{m_{1}+1}$.
Hence

$\mathcal{S}_{\psi}(n_{k_{1}})=\sum_{j=1}^{n_{k_{1}}}\psi^{\prime}(j)\chi_{\mathcal{S}}(j)=\sum_{j=1}^{m_{1}}\psi^{\prime}(v_{j})=\mathcal{S}_{\psi}(v_{m_{1}}),$

and so,

$\frac{\mathcal{S}_{\psi}(v_{m_{1}})}{\psi(v_{m_{1}})}=\frac{\mathcal{S}_{\psi}(n_{k_{1}})}{\psi(v_{m_{1}})}\geq\frac{\mathcal{S}_{\psi}(n_{k_{1}})}{\psi(n_{k_{1}})}>\overline{\operatorname{d}}_{\psi}(\mathcal{S})-\varepsilon.$

Thus, there exists at least one integer $m_{1}$ such that

$\displaystyle\frac{\mathcal{S}_{\psi}(v_{m_{k}})}{\psi(v_{m_{k}})}>\overline{\operatorname{d}}_{\psi}(\mathcal{S})-\varepsilon.$
(3.6)

For any such $m_{1}$, there exists an integer $n_{k_{2}}\geq v_{m_{1}+1}$
satisfying (3.5). Similarly, there exists an integer $m_{2}$ such that
$v_{m_{2}}\leq n_{k_{2}}<v_{m_{2}+1}$. Hence,
$\mathcal{S}_{\psi}(n_{k_{2}})=\mathcal{S}_{\psi}(v_{m_{2}})$ and

$\frac{\mathcal{S}_{\psi}(v_{m_{2}})}{\psi(v_{m_{2}})}=\frac{\mathcal{S}_{\psi}(n_{k_{2}})}{\psi(v_{m_{2}})}\geq\frac{\mathcal{S}_{\psi}(n_{k_{2}})}{\psi(n_{k_{2}})}>\overline{\operatorname{d}}_{\psi}(\mathcal{S})-\varepsilon.$

Therefore, $m_{2}$ satisfies (3.6). Note that since $n_{k_{2}}\geq
v_{m_{1}+1}$, we have

$\mathcal{S}_{\psi}(v_{m_{2}})=\mathcal{S}_{\psi}(n_{k_{2}})\geq\mathcal{S}_{\psi}(v_{m_{1}+1})>\mathcal{S}_{\psi}(v_{m_{1}}).$

Thus, there exist an infinite strictly increasing sequence $(m_{k})_{k}$ of
integers that satisfies (3.6), and consequently

$\limsup_{n\to\infty}\frac{\mathcal{S}_{\psi}(v_{n})}{\psi(v_{n})}\geq\overline{\operatorname{d}}_{\psi}(\mathcal{S}).$

This completes the proof of Theorem 3.1. ∎

Remark. From the proof of Theorem 3.1, we observe that the limit superior in
(3.2) remains valid for $\psi\in\mathcal{D}_{2}$ without the need for any
additional conditions.

######  Example 3.3.

We illustrate the usefulness of Theorem 3.1 in counting asymptotic densities.
Let $\mathcal{S}$ be the set of positive even integers. For
$\psi(x)=\sqrt{x}\in\mathcal{D}_{1}$, we may use Lemma 2.2 (or Example 2.3) to
obtain

$\displaystyle\mathcal{S}_{\psi}(2n)$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{2n}\frac{\chi_{\mathcal{S}}(k)}{2\sqrt{k}}=\sum_{k=1}^{n}\frac{1}{2\sqrt{2k}}=\frac{1}{\sqrt{2}}\sum_{k=1}^{n}\psi^{\prime}(k)\sim\frac{\psi(n)}{\sqrt{2}}=\frac{\sqrt{n}}{\sqrt{2}},\quad
n\to\infty.$

Therefore, by Theorem 3.1,

$\operatorname{d}_{\psi}(\mathcal{S})=\lim_{n\to\infty}\frac{\mathcal{S}_{\psi}(2n)}{\psi(2n)}=\frac{1}{2}.$

Consider now $\psi(x)=x\log x\in\mathcal{D}_{2}$. Noting that

$\frac{\psi^{\prime}(2n)}{\psi(n)}=\frac{\log(2n)+1}{n\log n}\to 0,\quad
n\to\infty,$

we have

$\mathcal{S}_{\psi}(2n)=\sum_{k=1}^{2n}(1+\log
k)\chi_{\mathcal{S}}(k)=\sum_{k=1}^{n}(1+\log(2k))=n(1+\log 2)+\log n!.$

Therefore, by Stirling’s formula and Theorem 3.1,

$\operatorname{d}_{\psi}(\mathcal{S})=\lim_{n\to\infty}\frac{\mathcal{S}_{\psi}(2n)}{\psi(2n)}=\lim_{n\to\infty}\frac{\log
n!}{2n\log(2n)}=\frac{1}{2}.$

Remark. It seems that the condition $\psi^{\prime}(v_{n})/\psi(n)\to 0$, as
$n\to\infty$, is sufficient but not necessary for the conclusion of Theorem
3.1. For example, consider the function
$\psi(x)=e^{\sqrt{x}}\in\mathcal{D}_{2}$ satisfying

$\frac{\psi^{\prime}(2n)}{\psi(n)}\to\infty,\quad n\to\infty.$

If $\mathcal{S}$ denotes the set of positive even integers, it follows that

$\mathcal{S}_{\psi}(2n+1)=\mathcal{S}_{\psi}(2n)=\sum_{k=1}^{2n}\frac{e^{\sqrt{k}}}{2\sqrt{k}}\chi_{\mathcal{S}}(k)=\sum_{k=1}^{n}\frac{e^{\sqrt{2k}}}{2\sqrt{2k}}\sim\frac{1}{2}e^{\sqrt{2n}},\quad\text{as
}n\to\infty,$

which yields

$\lim_{n\to\infty}\frac{\mathcal{S}_{\psi}(2n+1)}{\psi(2n+1)}=\frac{1}{2}\quad\text{and}\quad\lim_{n\to\infty}\frac{\mathcal{S}_{\psi}(2n)}{\psi(2n)}=\frac{1}{2}.$

Thus $\operatorname{d}_{\psi}(\mathcal{S})=1/2$.

##  4\. Regularity of asymptotic $\psi$-densities

In this section, we will use the notation $(u_{n})_{n}$ to represent an
arithmetic sequence of positive integers. For the sake of simplicity, these
sequences are defined as $u_{n}=an+b$, where $a\in\mathbb{N}$ and $b$ is a
non-negative integer. We denote $\mathcal{A}\subset\mathbb{N}$ as any subset
with elements forming an arithmetic progression of positive integers, formally
expressed as $\mathcal{A}=\\{u_{n}:n\in\mathbb{N}\\}$.

A function $\psi\in\mathcal{D}$ and the corresponding asymptotic
$\psi$-density $\operatorname{d}_{\psi}$ are said to be regular if
$\operatorname{d}_{\psi}(\mathcal{A})=1/a$ for any fixed integers $a$ and $b$.
This definition originates from a well-known fact [11] that both the
asymptotic density and the logarithmic density are regular. We arrive at the
following question.

######  Question 3.

What conditions on $\psi\in\mathcal{D}$ determine that $\psi$ is regular?

We will consider regularity in the classes $\mathcal{D}_{1}$ and
$\mathcal{D}_{2}$.

###  4.1. Regularity in $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$

As the main result of this subsection, we prove that all functions in
$\mathcal{D}_{1}$ are regular, while functions in $\mathcal{D}_{2}$ are
regular under certain additional conditions.

######  Theorem 4.1.

Let $(u_{n})_{n}$ be an infinite arithmetic sequence of positive integers.
Suppose that one of the following conditions holds:

  * (a)

$\psi\in\mathcal{D}_{1}$,

  * (b)

$\psi\in\mathcal{D}_{2}$ and $\psi^{\prime}(u_{n})/\psi(n)\to 0$, as
$n\to\infty$.

Then $\operatorname{d}_{\psi}(\mathcal{A})=1/a.$

###### Proof.

From Theorem 3.1, it suffices to show that

$\displaystyle\lim_{n\to\infty}\frac{\mathcal{A}_{\psi}(u_{n})}{\psi(u_{n})}=\frac{1}{a},$
(4.1)

where $u_{n}=an+b$ and

$\mathcal{A}_{\psi}(u_{n})=\sum_{k=1}^{u_{n}}\psi^{\prime}(k)\chi_{\mathcal{A}}(k)=\sum_{k=1}^{n}\psi^{\prime}(u_{k}).$

By the mean value theorem,

$\displaystyle\psi(u_{n})=$
$\displaystyle\big{(}\psi(u_{n})-\psi(u_{n-1})\big{)}+\big{(}\psi(u_{n-1})-\psi(u_{n-2})\big{)}+\cdots+\big{(}\psi(u_{2})-\psi(u_{1})\big{)}+\psi(u_{1})$
$\displaystyle=$ $\displaystyle
a\big{(}\psi^{\prime}(c_{n,n-1})+\psi^{\prime}(c_{n-1,n-2})+\cdots+\psi^{\prime}(c_{2,1})\big{)}+\psi(u_{1}),$

where $c_{k,k-1}\in(u_{k-1},u_{k})$ for $k=2,\ldots,n$.

  * (a)

If $\psi\in\mathcal{D}_{1}$, then the derivative $\psi^{\prime}$ is decreasing
and

$a\sum_{k=2}^{n}\psi^{\prime}(u_{k})+\psi(u_{1})\leq\psi(u_{n})\leq
a\sum_{k=1}^{n-1}\psi^{\prime}(u_{k})+\psi(u_{1}),$

that is,

$a\frac{\mathcal{A}_{\psi}(u_{n})}{\psi(u_{n})}+\frac{\psi(u_{1})-a\psi^{\prime}(u_{1})}{\psi(u_{n})}\leq
1\leq
a\frac{\mathcal{A}_{\psi}(u_{n})}{\psi(u_{n})}-\frac{a\psi^{\prime}(u_{n})-\psi(u_{1})}{\psi(u_{n})}.$
(4.2)

By the mean value theorem,

$0\leq\psi^{\prime}(x)\leq\psi(x)-\psi(x-1),\quad x>1,$

and hence

$0\leq\frac{\psi^{\prime}(x)}{\psi(x)}\cdot\frac{\psi(x)}{\psi(x-1)}\leq\frac{\psi(x)}{\psi(x-1)}-1,\quad
x>1.$

Using (2.3) and the squeeze theorem, we find that
$\frac{\psi^{\prime}(x)}{\psi(x)}\to 0$ as $x\to\infty$. Taking limit inferior
on the second inequality in (4.2) and limit superior on the first inequality
in (4.2) gives us

$\frac{1}{a}\leq\underline{\operatorname{d}}_{\psi}(\mathcal{A})\leq\overline{\operatorname{d}}_{\psi}(\mathcal{A})\leq\frac{1}{a},$

from which (4.1) follows immediately.

  * (b)

If $\psi\in\mathcal{D}_{2}$, then $\psi^{\prime}$ is increasing and

$a\mathcal{A}_{\psi}(u_{n})-a\psi^{\prime}(u_{n})+\psi(u_{1})\leq\psi(u_{n})\leq
a\mathcal{A}_{\psi}(u_{n})+\psi(u_{1})-a\psi^{\prime}(u_{1}).$

By making use of Theorem 3.1 and Lemma 2.4, and deducing similarly as in Case
(a), we obtain (4.1).

This completes the proof. ∎

###  4.2. More sufficient conditions for regularity in $\mathcal{D}_{2}$

As one may notice, for the condition

$\displaystyle\frac{\psi^{\prime}(u_{n})}{\psi(n)}=\frac{\psi^{\prime}(u_{n})}{\psi(u_{n})}\cdot\frac{\psi(u_{n})}{\psi(n)}\to
0,\quad n\to\infty,$ (4.3)

in Theorem 4.1 to hold for $\psi\in\mathcal{D}_{2}$, it suffices that
$\frac{\psi(u_{n})}{\psi(n)}$ is bounded, because we have
$\psi^{\prime}(u_{n})/\psi(u_{n})\to 0$ by Lemma 2.4. The following example
shows that the condition (4.3) may not always hold.

######  Example 4.2.

The test function $\psi(x)=x^{p}$, $p\geq 1$, does satisfy (4.3), but the
function $\psi(x)=\exp\big{(}x^{\sigma}\big{)}$, $\sigma\in(0,1)$, does not.
Indeed,

$u_{n}^{\sigma}-n^{\sigma}=\sigma\int_{n}^{u_{n}}t^{\sigma-1}\,dt\geq\sigma
u_{n}^{\sigma-1}(u_{n}-n),$

and hence

$\frac{\psi^{\prime}(u_{n})}{\psi(n)}=\sigma
u_{n}^{\sigma-1}\exp\big{(}u_{n}^{\sigma}-n^{\sigma}\big{)}\to\infty,\quad
n\to\infty,$

whenever $a\geq 2$ in $u_{n}=an+b$.

From Example 4.2, one may wonder if we have to restrict ourselves to functions
$\psi$ of finite order in the sense of

$\limsup_{x\to\infty}\frac{\log\psi(x)}{\log x}<\infty.$ (4.4)

However, (4.4) alone is not sufficient to guarantee that $\psi(u_{n})/\psi(n)$
is bounded. Indeed, in Lemma A.3 below we will construct a finite order
function $\psi$ for which $\psi(u_{n})/\psi(n)$ is unbounded and for which
$\psi(x+1)\not\sim\psi(x)$. Recall from Lemma 2.5 that $\psi(x+1)\sim\psi(x)$
for all $\psi\in\mathcal{D}_{2}$.

The discussion above leads to the following question.

######  Question 4.

What conditions on $\psi$ ensure the boundedness of $\psi(an+b)/\psi(n)$?

To explore this further, let us verify that $\psi$ needs to be of finite
order. Interestingly, it appears that the constant $b$ is irrelevant in this
context.

######  Lemma 4.3.

Let $\psi:(0,\infty)\to(0,\infty)$ be a strictly increasing function
satisfying

$\frac{\psi(ax)}{\psi(x)}\leq C<\infty$

for some integer $a\geq 2$. Then the order of $\psi$ is at most $\log C/\log
a$.

###### Proof.

Let $x\geq a$ be large. Then there exists an $N\in\mathbb{N}$ such that
$a^{N}\leq x\leq a^{N+1}$, that is,

$N\leq\frac{\log x}{\log a}\leq N+1\quad\textnormal{and}\quad
1\leq\frac{x}{a^{N}}\leq a.$

Deducing inductively,

$\displaystyle\psi(ax)$ $\displaystyle\leq$ $\displaystyle C\psi(x)\leq
C^{2}\psi\left(\frac{x}{a}\right)\leq\cdots\leq
C^{N+1}\psi\left(\frac{x}{a^{N}}\right)$ $\displaystyle\leq$ $\displaystyle
C\psi(a)\exp\big{(}N\log C\big{)}\leq C\psi(a)x^{\frac{\log C}{\log a}}.$

Thus $\psi(x)=O\left(x^{\frac{\log C}{\log a}}\right)$, from which the
assertion follows. ∎

Remark. Functions $\psi\in\mathcal{D}_{2}$ are not in general of finite order,
but they do have a growth restriction $\log\psi(x)=o(x)$. To see this, we
first recall from Lemma 2.5 that $\psi(x+1)\sim\psi(x)$ as $x\to\infty$. The
assertion then follows by Lemma A.2 below.

Based on Example 4.2 and Lemma 4.3, a natural question is, how strong the
asymptotic equation $\psi(x+1)\sim\psi(x)$ is for a function
$\psi\in\mathcal{D}_{2}$? Let us consider the test function $\psi(x)=x^{p}$,
$p\geq 1$, for which

$\frac{\psi(x+1)}{\psi(x)}=\left(1+\frac{1}{x}\right)^{p}.$

Consequently,

$\limsup_{x\to\infty}\frac{\log\frac{\psi(x+1)}{\psi(x)}}{\log\left(1+1/x\right)}<\infty.$
(4.5)

This condition can be simplified by using L’Hospital’s rule, which gives us

$\lim_{x\to\infty}\frac{\log\left(1+1/x\right)}{1/x}=\lim_{y\to
0^{+}}\frac{\log(1+y)}{y}=1.$

Thus (4.5) is equivalent to

$\limsup_{x\to\infty}\left(x\log\frac{\psi(x+1)}{\psi(x)}\right)<\infty.$
(4.6)

This condition is satisfied by $\psi(x)=x^{p}$, but in general it illustrates
the strength of the asymptotic equation $\psi(x+1)\sim\psi(x)$. We are mostly
interested in unbounded functions $\psi$, but an example of a bounded function
$\psi$ satisfying both $\psi(x+1)\sim\psi(x)$ and (4.6) is given by
$\psi(x)=\arctan(x)$.

Condition (4.6) gives an answer to Question 4, as stated in Theorem 4.4 below.
It is worth noting that neither convexity nor differentiability is necessary
for this result.

######  Theorem 4.4.

Let $\psi:(0,\infty)\to(1,\infty)$ be a function. Then the following
assertions hold.

  * (a)

If (4.6) holds, then (4.4) holds and the ratio $\frac{\psi(an+b)}{\psi(n)}$ is
a bounded function of $n$ for any fixed constants $a,b\in\mathbb{N}$.

  * (b)

If $\displaystyle\liminf_{x\to\infty}\left(\frac{x}{\log
x}\log\frac{\psi(x+1)}{\psi(x)}\right)>0$, then
$\displaystyle\liminf_{x\to\infty}\frac{\log\psi(x)}{\log x}=\infty$ and the
ratio $\frac{\psi(ax)}{\psi(x)}$ is an unbounded function of $x$ for any fixed
constant $a>1$.

###### Proof.

(a) Note that both assertions are clearly true if $\psi$ is bounded. Hence we
may assume that $\psi$ is unbounded. Let $c=\max\\{a,b\\}$. By (4.6), there
exist constants $K>0$ and $R>0$ such that

$\frac{\psi(x+1)}{\psi(x)}\leq\exp\left(\frac{K}{x}\right),\quad x>R.$ (4.7)

Then

$\displaystyle\frac{\psi(an+b)}{\psi(n)}$ $\displaystyle\leq$
$\displaystyle\frac{\psi(c(n+1))}{\psi(n)}=\frac{\psi(c(n+1))}{\psi(c(n+1)-1)}\cdot\frac{\psi(c(n+1)-1)}{\psi(c(n+1)-2)}\cdots\frac{\psi(n+1)}{\psi(n)}$
$\displaystyle\leq$
$\displaystyle\exp\left(K\sum_{k=n}^{c(n+1)-1}\frac{1}{k}\right)\leq\exp\left(\frac{Kc(n+1)}{n}\right)\leq
e^{2Kc},\quad n>R,$

which proves the second assertion in (a). We now proceed to prove the first
assertion in (a). To simplify the reasoning, we may suppose, without loss of
generality, that the consequence (4.7) of (4.6) is valid for all $x>0$. Let
$x\geq 2$ be large. Then there exists an $N\in\mathbb{N}$ such that $N+1\leq
x\leq N+2$. Deducing inductively,

$\displaystyle\psi(x+1)$ $\displaystyle\leq$
$\displaystyle\psi(x)\exp\left(\frac{K}{x}\right)\leq\psi(x-1)\exp\left(\frac{K}{x-1}+\frac{K}{x}\right)\leq\cdots$
$\displaystyle\leq$
$\displaystyle\psi(x-N)\exp\left(K\sum_{j=0}^{N}\frac{1}{x-j}\right)\leq\psi(2)\exp\left(K\sum_{j=1}^{N+1}\frac{1}{x-(j-1)}\right)$
$\displaystyle\leq$
$\displaystyle\psi(2)\exp\left(K\sum_{j=1}^{N+1}\frac{1}{j}\right)\leq\psi(2)\exp\big{(}O(\log
x)\big{)},$

which implies the first assertion in (a).

(b) By the assumption, there exist constants $\delta>0$ and $R>0$ such that

$\frac{\psi(x+1)}{\psi(x)}\geq\exp\left(\frac{\delta\log x}{x}\right),\quad
x>R.$

Recall that the function $x\mapsto\frac{\log x}{x}$ is decreasing for $x\geq
e$. Hence, deducing similarly as in (a), but for a large $x$ satisfying
$N+3\leq x\leq N+4$, we obtain

$\psi(x+1)\geq\psi(3)\exp\left(O\left(\log^{2}x\right)\right),$

proving the first assertion in (b). The second assertion in (b) follows by
Lemma 4.3 because $\psi$ is of infinite order. ∎

Remark. The assumption in Theorem 4.4(b) can be replaced with a weaker
assumption

$\liminf_{x\to\infty}\left(\frac{x}{h(x)}\log\frac{\psi(x+1)}{\psi(x)}\right)>0,$

where $h$ is a continuous function tending to infinity arbitrarily slowly such
that $\frac{h(x)}{x}$ is decreasing. Indeed, we obtain, for some $\delta>0$
and for $N+1\leq x\leq N+2$,

$\displaystyle\psi(x+1)$ $\displaystyle\geq$
$\displaystyle\psi(x)\exp\left(\frac{\delta
h(x)}{x}\right)\geq\psi(x-1)\exp\left(\frac{\delta h(x)}{x}+\frac{\delta
h(x-1)}{x-1}\right)$ $\displaystyle\geq$
$\displaystyle\ldots\geq\psi(x-N)\exp\left(\delta\sum_{j=0}^{N}\frac{h(x-j)}{x-j}\right).$

Consequently, using the fact that $\frac{h(x)}{x}$ is decreasing,

$\log\psi(x+1)\geq O(1)+\delta\int_{\alpha}^{x+1}\frac{h(t)}{t}\,dt,$

where $\alpha>0$ is a number independent of $x$. By L’Hospital’s rule,

$\frac{\int_{\alpha}^{x+1}\frac{h(t)}{t}\,dt}{\log(x+1)}\sim
h(x+1)\to\infty,\quad x\to\infty,$

and we conclude that $\psi$ is of infinite order. The fact that
$\frac{\psi(ax)}{\psi(x)}$ is an unbounded function of $x$ follows again by
Lemma 4.3.

##  Appendix A On the identity $\psi(n+1)\sim\psi(n)$

In this appendix, we explore the possibility of convex functions $\psi$ to
satisfy the crucial identity (2.3). We will show that a convex function $\psi$
of finite order, as a restriction, does not always satisfy (2.3). Recall (4.4)
for the definition of order. This appendix can be of independent interest to
readers interested in the properties of convex functions.

Before we proceeding, let us provide an illustrative example of convex
functions that do satisfy (2.3).

######  Example A.1.

Functions $\psi(x)=x^{p}$, $p\geq 1$, satisfy (2.3). This follows from

$px^{p-1}\leq(x+1)^{p}-x^{p}=p\int_{x}^{x+1}t^{p-1}\,dt\leq p(x+1)^{p-1}.$

On the other hand, if $p\in(0,1)$, then

$p(x+1)^{p-1}\leq(x+1)^{p}-x^{p}=p\int_{x}^{x+1}t^{p-1}\,dt\leq px^{p-1},$

and we see that $\psi(x)=\exp\big{(}x^{p}\big{)}$ satisfies (2.3).

A strictly increasing and measurable (or continuous) function
$\psi:(r_{0},\infty)\to(1,\infty)$ is called _additively slowly varying_ (or
translational slowly varying) [16], if

$\psi(x+\lambda)\sim\psi(x),\quad x\to\infty,$ (A.1)

for all $\lambda>0$. This is a slightly stronger condition than (2.3). Next we
prove that (A.1) (as well as (2.3)) implies a growth restriction for $\psi$,
keeping in mind that $\psi(x)=e^{x}$ does not satisfy either of (2.3) or
(A.1).

######  Lemma A.2.

If $\psi:(r_{0},\infty)\to(1,\infty)$ is additively slowly varying, then

$\log\psi(x)=o(x).$ (A.2)

###### Proof.

Let $\varepsilon>0$ and $\lambda>0$. Then there exists a constant $R>r_{0}$
such that

$\psi(x+\lambda)\leq(1+\varepsilon)\psi(x),\quad x\geq R.$

Let $x\geq R+\lambda$. Then there exists an $N\in\mathbb{N}$ such that
$R+N\lambda\leq x\leq R+(N+1)\lambda$. Deducing inductively, we obtain

$\psi(x+\lambda)\leq(1+\varepsilon)^{N+1}\psi(x-N\lambda)\leq(1+\varepsilon)^{N+1}\psi(R+\lambda).$

Since $N\leq\frac{x-R}{\lambda}$, we obtain

$\log\psi(x)\leq\frac{\log(1+\varepsilon)}{\lambda}x+O(1),\quad x\geq
R+2\lambda.$

Consequently,
$\limsup\limits_{x\to\infty}\frac{\log\psi(x)}{x}\leq\frac{\log(1+\varepsilon)}{\lambda}$,
where we may let $\varepsilon\to 0^{+}$. ∎

Conversely to Lemma A.2 (keeping in mind that we are currently interested in
convex functions), we may ask if an increasing, continuous and convex function
$\psi:(r_{0},\infty)\to(1,\infty)$ satisfying a growth restriction would
always satisfy (2.3). Apparently, this is not the case.

######  Lemma A.3.

For every $p>1$ there exist a strictly increasing and unbounded sequence
$(x_{n})_{n}$ of positive real numbers and a strictly increasing, continuous
and convex function $\psi:[x_{1},\infty)\to(1,\infty)$ that is linear on each
interval $[x_{n},x_{n+1}]$ and satisfies

$\lim_{n\to\infty}\frac{\psi(x_{n}+1)}{\psi(x_{n})}=\left\\{\begin{array}[]{rl}1,&1<p<2,\\\
2,&p=2,\\\ \infty,&p>2.\end{array}\right.$ (A.3)

Moreover, the function $\psi$ has the growth rate

$\lim_{x\to\infty}\frac{\log\psi(x)}{\log x}=p,$ (A.4)

that is, the limit in (A.4) exists and is equal to $p$.

###### Proof.

We generalize the reasoning used in constructing [8, Example 1]. Let
$x_{n}=2^{p^{n}}$ for $n\in\mathbb{N}$, so that $x_{n+1}=x_{n}^{p}$. We let
$\psi$ to be the piecewise linear function inscribed in $x^{p}$, that is,
$\psi(x_{n})=x_{n}^{p}$ and $\psi$ is linear on the intervals
$[x_{n},x_{n+1}]$ with

$\begin{split}\psi(x)&=\psi(x_{n})+\frac{\psi(x_{n+1})-\psi(x_{n})}{x_{n+1}-x_{n}}(x-x_{n})\\\
&=x_{n}^{p}+\frac{x_{n+1}^{p}-x_{n}^{p}}{x_{n+1}-x_{n}}(x-x_{n}),\quad
x\in[x_{n},x_{n+1}].\end{split}$ (A.5)

It is obvious that $\psi$ is strictly increasing, continuous and convex.
Moreover,

$\frac{\psi(x_{n}+1)}{\psi(x_{n})}=1+\frac{x_{n+1}^{p}-x_{n}^{p}}{x_{n}^{p}(x_{n+1}-x_{n})}\sim
1+x_{n}^{p(p-2)}\to\left\\{\begin{array}[]{rl}1,&1<p<2,\\\ 2,&p=2,\\\
\infty,&p>2.\end{array}\right.$

This proves (A.3). It remains to prove (A.4). To this end, let
$\alpha\in[0,1]$, and denote $y_{n}=\alpha x_{n}+(1-\alpha)x_{n+1}$. Then

$\psi(y_{n})=x_{n}^{p}+(1-\alpha)(x_{n+1}^{p}-x_{n}^{p})=(1-\alpha)x_{n+1}^{p}+\alpha
x_{n}^{p}.$

We have

$\frac{\log\psi(x_{n})}{\log x_{n}}=p,\quad n\in\mathbb{N},$

and

$\frac{\log\psi(y_{n})}{\log
y_{n}}\sim\frac{\log\big{(}(1-\alpha)x_{n+1}^{p}\big{)}}{\log\big{(}(1-\alpha)x_{n+1}\big{)}}\to
p,\quad n\in\mathbb{N},\ \alpha\in(0,1).$

For any real $x\geq x_{1}$ there exists an $N\in\mathbb{N}$ such that
$x_{N}\leq x\leq x_{N+1}$. In other words, $x=y_{N}$ for some
$\alpha\in[0,1]$. The assertion (A.4) follows by the continuity of $\psi$. ∎

The growth condition (A.2) is of special interest as it is not satisfied by
$\psi(x)=e^{x}$. Next, in addition to (A.2), we assume that $\log\psi$ is
concave, that is, $\psi$ is log-concave. Note that the asymptotic identity
(2.2), which is satisfied by the functions $\psi\in\mathcal{D}_{2}$ by
definition, is not assumed. See also Lemma 2.5 for comparison.

######  Lemma A.4.

Let $\psi:(r_{0},\infty)\to(1,\infty)$ be a strictly increasing,
differentiable, unbounded and convex function. In addition, we suppose that
$\psi$ is log-concave, and that (A.2) holds. Then (2.3) holds.

###### Proof.

By differentiability and log-concavity, the derivative

$(\log\psi(x))^{\prime}=\frac{\psi^{\prime}(x)}{\psi(x)}$

is a decreasing function. Suppose that there are constants $\delta>0$ and
$R>0$ such that

$\frac{\psi^{\prime}(t)}{\psi(t)}>\delta,\quad t\geq R.$

Integrating both sides over the interval $[R,x]$, we find that

$\log\psi(x)-\log\psi(R)\geq\delta(x-R),\quad x\geq R,$

which violates the assumption (A.2). Therefore, as a decreasing and positive
function, the logarithmic derivative $\psi^{\prime}(x)/\psi(x)$ tends to zero
as $x\to\infty$. The assertion then follows by Lemma 2.4. ∎

Remark. (a) If $\psi$ is twice differentiable, then log-concavity can be
expressed equivalently as

$(\log\psi(x))^{\prime\prime}=\left(\frac{\psi^{\prime}(x)}{\psi(x)}\right)^{\prime}=\frac{\psi^{\prime\prime}(x)}{\psi(x)}-\left(\frac{\psi^{\prime}(x)}{\psi(x)}\right)^{2}\leq
0.$

Writing this as $\psi^{\prime\prime}/\psi^{\prime}\leq\psi^{\prime}/\psi$ and
integrating over the interval $[a,x]$, we obtain

$\log\frac{\psi^{\prime}(x)}{\psi^{\prime}(a)}\leq\log\frac{\psi(x)}{\psi(a)}\quad\Longrightarrow\quad\frac{\psi^{\prime}(x)}{\psi(x)}\leq\frac{\psi^{\prime}(a)}{\psi(a)}.$

Integrating again results in $\log\psi(x)=O(x)$. This is a weaker growth
restriction than (A.2), and is satisfied by $\psi(x)=e^{x}$. Hence (A.2) needs
to be assumed in Lemma A.4 even if $\psi$ would be twice differentiable.

(b) Let $\psi$ be the convex function in Lemma A.3, and let $C>0$ be a large
constant. Then there exists an $N\in\mathbb{N}$ such that $x_{n}+C<x_{n+1}$
for all $n\geq N$. From (A.5),

$\frac{\psi^{\prime}(x)}{\psi(x)}\geq\frac{x_{n+1}^{p}-x_{n}^{p}}{\psi(x_{n}+C)(x_{n+1}-x_{n})}\sim\frac{1}{C},\quad
x_{n}<x<x_{n}+C,$ (A.6)

for all $n$ large enough. This means that the logarithmic derivative
$\psi^{\prime}/\psi$ does not tend to zero as $x\to\infty$. Further, let
$y_{n}=\frac{x_{n}+x_{n+1}}{2}$. Then, from (A.5),

$\frac{\psi^{\prime}(y_{n})}{\psi(y_{n})}=\frac{2(x_{n+1}^{p}-x_{n}^{p})}{(x_{n+1}^{p}+x_{n}^{p})(x_{n+1}-x_{n})}\sim\frac{2}{x_{n+1}}\to
0,\quad n\to\infty.$

Combining this with (A.6), we find that $\psi^{\prime}(x)/\psi(x)$ is not a
decreasing function, which means that $\psi$ is not log-concave. In
particular, differentiability (outside of a discrete set) together with a
growth restriction such as (A.2) or (A.4) does not imply log-concavity.
Further, the definition of the function $\psi$ can be slightly modified by
smoothing the vertices in the graph of $\psi$ in such a way that $\psi$
becomes differentiable on $(x_{1},\infty)$. This procedure has no affect on
the properties of $\psi$ described in this remark.

Finally, having seen that a differentiable function $\psi$ satisfying the
growth restriction (A.2) is not always log-concave, and its logarithmic
derivative $\psi^{\prime}/\psi$ does not always tend to zero as $x\to\infty$,
it is natural to ask what is the size of the set

$E:=\left\\{x>r_{0}:\frac{\psi^{\prime}(x)}{\psi(x)}>\varepsilon\right\\},$
(A.7)

where $\varepsilon>0$ is a fixed small constant? The following lemma answers
this question by means of the continuous asymptotic density, that is,
$\psi(x)=x$ in (1.7).

######  Lemma A.5.

Let $\psi:(r_{0},\infty)\to(1,\infty)$ be a strictly increasing,
differentiable and unbounded function satisfying (A.2), and let
$\varepsilon>0$ be an arbitrarily small constant. Then the set $E$ in (A.7)
has an asymptotic density $\operatorname{dens}(E)=0$.

###### Proof.

By (A.2) and (A.7),

$\int_{E\cap(r_{0},x)}dt\leq\frac{1}{\varepsilon}\int_{r_{0}}^{x}\frac{\psi^{\prime}(t)}{\psi(t)}\,dt\leq\frac{1}{\varepsilon}\left(\log\psi(x)+O(1)\right)=o(x),$

and the assertion follows. ∎

##  Appendix B Analytic and Abel densities

Recalling that

$\int_{1}^{\infty}\frac{dx}{x^{p}}=\frac{1}{p-1},\quad p>1,$

the _lower and upper analytic densities_ of a set $A\subset\mathbb{N}$ can be
expressed, respectively, by

$\underline{\operatorname{d}}_{a}(A)=\liminf_{p\to
1^{+}}(p-1)\sum_{n=1}^{\infty}\frac{\chi_{A}(n)}{n^{p}}\quad\textnormal{and}\quad\overline{\operatorname{d}}_{a}(A)=\limsup_{p\to
1^{+}}(p-1)\sum_{n=1}^{\infty}\frac{\chi_{A}(n)}{n^{p}},$

see Section 2.3. If
$\underline{\operatorname{d}}_{a}(A)=\overline{\operatorname{d}}_{a}(A)$, then
their common value $\operatorname{d}_{a}(A)$ is called the _analytic density_
of $A$ discussed in Section 2.3. We have seen in Theorem 2.8 that for any
$A\subset\mathbb{N}$, the analytic density $\operatorname{d}_{a}(A)$ exists if
and only if the logarithmic density $\operatorname{\delta}(A)$ exists. The
technique used in the proof of Theorem 2.8, see [18, p. 418], can be modified
to prove Lemma B.1 below. The lemma shows that the existence of the
logarithmic density of $A\subset\mathbb{N}$ implies the existence of the
analytic density of $A$, thus giving an alternative proof of one direction of
Theorem 2.8.

######  Lemma B.1.

If $A\subset\mathbb{N}$, then

$0\leq\underline{\operatorname{\delta}}(A)\leq\underline{\operatorname{d}}_{a}(A)\leq\overline{\operatorname{d}}_{a}(A)\leq\overline{\operatorname{\delta}}(A)\leq
1.$ (B.1)

###### Proof.

It suffices to prove that

$\underline{\operatorname{\delta}}(A)\leq\underline{\operatorname{d}}_{a}(A)\quad\textnormal{and}\quad\overline{\operatorname{d}}_{a}(A)\leq\overline{\operatorname{\delta}}(A).$

The first inequality is trivial if $\underline{\operatorname{\delta}}(A)=0$,
so we suppose that $\underline{\operatorname{\delta}}(A)>0$. The proof of the
second inequality above does not depend on this restriction.

Let $\varepsilon\in(0,\underline{\operatorname{\delta}}(A))$. Then there
exists a $t_{0}=t_{0}(\varepsilon)>0$ such that

$\underline{\operatorname{\delta}}(A)-\varepsilon\leq\frac{A_{\log}(t)}{\log
t}\leq\overline{\operatorname{\delta}}(A)+\varepsilon,\quad t\geq t_{0},$

where

$A_{\log}(x)=\begin{cases}\sum_{1\leq k\leq x}\frac{\chi_{A}(k)}{k},&\text{if
}x\geq 1,\\\ 0,&\text{if }0\leq x<1.\end{cases}$

For any $x\geq 0$ there exists a unique non-negative integer $N$ such that
$N\leq x<N+1$. Then, by (1.9),

$A_{\log}(x)=A_{\log}(N)\leq\sum_{k=1}^{N}\frac{1}{k}\sim\log N\leq\log x.$

Using Riemann-Stieltjes integration and integration by parts on the step-
function $A_{\log}(n)$, we obtain

$\begin{split}\sum_{n=1}^{\infty}\frac{\chi_{A}(n)}{n^{p}}&=\sum_{n=1}^{\infty}\frac{\chi_{A}(n)}{n^{p-1}n}=\sum_{n=1}^{\infty}\frac{A_{\log}(n)-A_{\log}(n-1)}{n^{p-1}}=\int_{1}^{\infty}\frac{dA_{\log}(t)}{t^{p-1}}\\\
&=\left[\frac{A_{\log}(t)}{t^{p-1}}\right]_{1}^{\infty}+(p-1)\int_{1}^{\infty}\frac{A_{\log}(t)}{t^{p}}\,dt\\\
&=O(1)+(p-1)\int_{1}^{\infty}\frac{A_{\log}(t)}{\log t}\cdot\frac{\log
t}{t^{p}}\,dt,\quad p>1,\end{split}$ (B.2)

see [2, Theorem 7.9] and [2, Theorem 7.11]. Thus, using

$\int_{1}^{\infty}\frac{\log t}{t^{p}}\,dt=\frac{1}{(p-1)^{2}},\quad p>1,$

we obtain

$\underline{\operatorname{\delta}}(A)-\varepsilon+O(1)(p-1)\leq(p-1)\sum_{n=1}^{\infty}\frac{\chi_{A}(n)}{n^{p}}\leq\overline{\operatorname{\delta}}(A)+\varepsilon+O(1)(p-1).$

Taking limit inferior as $p\to 1^{+}$ on the first inequality and limit
superior as $p\to 1^{+}$ on the second inequality gives us

$\underline{\operatorname{\delta}}(A)-\varepsilon\leq\underline{\operatorname{d}}_{a}(A)\quad\textnormal{and}\quad\overline{\operatorname{d}}_{a}(A)\leq\overline{\operatorname{\delta}}(A)+\varepsilon.$

The assertions follow by letting $\varepsilon\to 0^{+}$. ∎

The other direction of Theorem 2.8 can be proved by using the following
Karamata’s theorem, see [18, p. 418] for details. Karamata’s theorem will be
needed later on for another purpose in this appendix.

######  Theorem B.2 ([18, p. 322]).

Let $A(t)$ be an increasing function such that the integral

$F(\sigma):=\int_{0}^{\infty}\mathrm{e}^{-\sigma t}dA(t)$

converges for all $\sigma>0$. Assume there exist two real numbers $c\geqslant
0$, $\omega>0$, such that

$F(\sigma)=\frac{c+o(1)}{\sigma^{\omega}},\quad\sigma\rightarrow 0+.$

Then we have

$A(x)=\frac{(c+o(1))x^{\omega}}{\Gamma(\omega+1)},\quad x\rightarrow+\infty.$

Recalling that

$\sum_{n=1}^{\infty}x^{n}=\frac{x}{1-x},\quad 0<x<1,$

the _lower and upper Abel densities_ of a set $A\subset\mathbb{N}$ are
defined, respectively, by

$\underline{\operatorname{d}}_{\text{Abel}}(A)=\liminf_{x\to
1^{-}}(1-x)\sum_{n=1}^{\infty}x^{n}\chi_{A}(n)\quad\textnormal{and}\quad\overline{\operatorname{d}}_{\text{Abel}}(A)=\limsup_{x\to
1^{-}}(1-x)\sum_{n=1}^{\infty}x^{n}\chi_{A}(n).$

If
$\underline{\operatorname{d}}_{\text{Abel}}(A)=\overline{\operatorname{d}}_{\text{Abel}}(A)$,
then their common value $\operatorname{d}_{\text{Abel}}(A)$ is called the
_Abel density_ of $A$ discussed in [15]. It might not be obvious at first
glance, but the Abel density is closely related to the asymptotic (linear)
density, as the following theorem illustrates.

######  Theorem B.3 ([15, Theorem 4.3]).

Let $A\subset\mathbb{N}$. Then the Abel density of $A$ exists if and only if
the asymptotic density of $A$ exists. In this case, the two densities are
equal.

The technique used in proving Lemma B.1 can be used to obtain a short
alternative proof for Lemma B.4 below. The lemma shows that the existence of
the asymptotic density of $A\subset\mathbb{N}$ implies the existence of the
Abel density of $A$, thus proving one direction of Theorem B.3.

######  Lemma B.4.

[15, Theorem 4.2] If $A\subset\mathbb{N}$, then

$\displaystyle
0\leq\underline{\operatorname{d}}(A)\leq\underline{\operatorname{d}}_{\textnormal{Abel}}(A)\leq\overline{\operatorname{d}}_{\textnormal{Abel}}(A)\leq\overline{\operatorname{d}}(A)\leq
1.$ (B.3)

###### Proof.

It suffices to prove that

$\underline{\operatorname{d}}(A)\leq\underline{\operatorname{d}}_{\textnormal{Abel}}(A)\quad\textnormal{and}\quad\overline{\operatorname{d}}_{\textnormal{Abel}}(A)\leq\overline{\operatorname{d}}(A),$

where we may again suppose that $\underline{\operatorname{d}}(A)>0$. The proof
of the second inequality above does not depend on this restriction.

We begin by noting that

$\int_{1}^{\infty}tx^{t}\,dt=\frac{(1-\log x)x}{\log^{2}x},\quad 0<x<1.$

This gives, in particular,

$\lim_{t\to\infty}tx^{t}=0,\quad 0<x<1,$

which can also be verified independently by L’Hospital’s rule. Let
$\varepsilon\in(0,\underline{\operatorname{d}}(A))$. Then there exists a
$t_{0}=t_{0}(\varepsilon)>0$ such that

$\underline{\operatorname{d}}(A)-\varepsilon\leq\frac{A(t)}{t}\leq\overline{\operatorname{d}}(A)+\varepsilon,\quad
t\geq t_{0},$

where

$A(x)=\begin{cases}\sum_{1\leq k\leq x}\chi_{A}(k),&\text{if }x\geq 1,\\\
0,&\text{if }0\leq x<1.\end{cases}$

It is clear that $A(x)\leq x$ for $x\geq 0$. Using Riemann-Stieltjes
integration and integration by parts on the step-function $A(n)$, we obtain

$\begin{split}\sum_{n=1}^{\infty}x^{n}\chi_{A}(n)&=\sum_{n=1}^{\infty}x^{n}\big{(}A(n)-A(n-1)\big{)}\\\
&=\int_{1}^{\infty}x^{t}dA(t)=\left[x^{t}A(t)\right]_{1}^{\infty}-\log
x\int_{1}^{\infty}x^{t}A(t)\,dt\\\
&=O(1)+\log\frac{1}{x}\int_{1}^{\infty}tx^{t}\cdot\frac{A(t)}{t}\,dt,\quad
0<x<1.\end{split}$ (B.4)

Thus

$\left(\underline{\operatorname{d}}(A)-\varepsilon\right)\frac{(1-\log
x)x}{-\log
x}+O(1)\leq\sum_{n=1}^{\infty}x^{n}\chi_{A}(n)\leq\left(\overline{\operatorname{d}}(A)+\varepsilon\right)\frac{(1-\log
x)x}{-\log x}+O(1).$

Since

$\lim_{x\to 1^{-}}(1-x)\frac{(1-\log x)x}{-\log x}=1,$

we find that there exists a $\delta\in(0,1)$ such that

$\underline{\operatorname{d}}(A)-2\varepsilon\leq(1-x)\sum_{n=1}^{\infty}x^{n}\chi_{A}(n)\leq\overline{\operatorname{d}}(A)+2\varepsilon,\quad
1-\delta<x<1.$

Taking limit inferior as $x\to 1^{-}$ on the first inequality and limit
superior as $x\to 1^{-}$ on the second inequality gives us

$\underline{\operatorname{d}}(A)-2\varepsilon\leq\underline{\operatorname{d}}_{\textnormal{Abel}}(A)\quad\textnormal{and}\quad\overline{\operatorname{d}}_{\textnormal{Abel}}(A)\leq\overline{\operatorname{d}}(A)+2\varepsilon.$

The assertion follows by letting $\varepsilon\to 0^{+}$. ∎

We give a new proof for the other direction of Theorem B.3 based on Karamata’s
theorem.

######  Lemma B.5.

Let $A\subset\mathbb{N}$. If $\delta:=\operatorname{d}_{\textnormal{Abel}}(A)$
exists, then $\operatorname{d}(A)$ exists and $\operatorname{d}(A)=\delta$.

###### Proof.

By the assumption, we have

$(1-x)\sum_{n=1}^{\infty}x^{n}\chi_{A}(n)=\delta+o(1),\quad x\to 1^{-}.$ (B.5)

Moreover, from (B.4),

$\sum_{n=1}^{\infty}x^{n}\chi_{A}(n)=\int_{0}^{\infty}x^{t}dA(t)=\int_{0}^{\infty}e^{-t\log\frac{1}{x}}dA(t).$

Recall that

$1-x<\log\frac{1}{x}<\frac{1-x}{x},\quad 0<x<1.$

Set $\sigma=\log\frac{1}{x}$, and define

$F(\sigma):=\int_{0}^{\infty}e^{-\sigma t}dA(t),\quad\sigma>0.$

Then, by (B.5),

$F(\sigma)=\frac{\delta+o(1)}{\sigma},\quad\sigma\to 0^{+}.$

By Theorem B.2, we get

$A(n)=(\delta+o(1))n,\quad n\to\infty.$

This completes the proof. ∎

Finally, we note that relations between upper and lower asymptotic and Abel
densities can be found in [5].

## References

  * [1] Alexander R., _Density and multiplicative structure of sets of integers_. Acta Arith. 12 (1966/67), 321–332.
  * [2] Apostol T. M., _Mathematical Analysis_. Second edition. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1974.
  * [3] Barry P. D., _The minimum modulus of small integral and subharmonic functions_. Proc. Lond. Math. Soc. (3) 12 (1962), 445–495.
  * [4] Cargo G. T., _Comparable means and generalized convexity_. J. Math. Anal. Appl. 12 (1965), 387–392.
  * [5] Filip F., A. Jankov and J. Šustek, _On relation between asymptotic and Abel densities._ J. Number Theory 209 (2020), 451–466.
  * [6] Grekos G., _On various definitions of density (survey)._ Tatra Mt. Math. Publ. 31 (2005), 17–27.
  * [7] Halberstam H. and K. F. Roth, _Sequences_. Springer-Verlag, New York-Heidelberg-Berlin, 1983.
  * [8] Janković S. and T. Ostrogorski, _Convex additively slowly varying functions._ J. Math. Anal. Appl. 274 (2002), no. 1, 228–238.
  * [9] Massé B. and D. Schneider, _A survey on weighted densities and their connection with the first digit phenomenon_. Rocky Mt. J. Math. 41 (2011), no. 5, 1395–1415.
  * [10] Niven I., H. S. Zuckerman and H. L. Montgomery, _An Introduction to the Theory of Numbers_. Fifth edition. John Wiley & Sons, Inc., New York, 1991.
  * [11] Pekara G. C., _The Asymptotic Density of Certain Integer Sequences_. PhD thesis, Oklahoma State University, 1972.
  * [12] Powell B. J. and T. Šalát, _Convergence of subseries of the harmonic series and asymptotic densities of sets of positive integers_. Publ. Inst. Math., Nouv. Sér. 50(64) (1991), 60–70.
  * [13] Rajagopal C. T., _Some limit theorems._ Amer. J. Math. 70 (1948), 157–166.
  * [14] Rohrbach H. and B. Volkmann, _Verallgemeinerte asymptotische dichten_. J. Reine Angew. Math. 194 (1955), 195–209.
  * [15] Sonnenschein D., _A General Theory of Asymptotic Density._ PhD thesis, University of British Columbia, 1971.
  * [16] Tasković M. R., _Survey on translational regularly varying functions_. Mathematica 7 (2003), 153–174.
  * [17] Taylor A. E., _L’Hospital’s rule._ Amer. Math. Monthly 59 (1952), 20–24.
  * [18] Tenenbaum G., _Introduction to Analytic and Probabilistic Number Theory._ Translated from the second French edition (1995) by C. B. Thomas. Cambridge Studies in Advanced Mathematics, 46. Cambridge University Press, Cambridge, 1995.

E-mail<EMAIL_ADDRESS>
University of Eastern Finland, Department of Physics and Mathematics, P.O. Box
111, 80100 Joensuu, Finland

E-mail<EMAIL_ADDRESS>
National Higher School of Mathematics, Scientific and Technology Hub of Sidi
Abdellah, P.O. Box 75, Algiers 16093, Algeria


# Evidence for the rare decay $\mathchar 28934\relax^{+}\rightarrow
p\mu^{+}\mu^{-}$ at LHCb

Francesco Dettori  CERN European Organization for Nuclear Research, CH1211
Geneve 23, Switzerland<EMAIL_ADDRESS>
###### Abstract

A search for the rare decay $\mathchar 28934\relax^{+}\rightarrow
p\mu^{+}\mu^{-}$ is performed using $pp$ collision data recorded by the LHCb
experiment at centre-of-mass energies $\sqrt{s}=7$ and
$8\mathrm{\,Te\kern-1.00006ptV}$, corresponding to an integrated luminosity of
$3\mbox{\,fb}^{-1}$. An excess of events is observed with respect to the
background expectations with a signal significance of $4.0$ standard
deviations. No significant structure is observed in the dimuon invariant mass
distribution.

##  1 Introduction

The $\mathchar 28934\relax^{+}\rightarrow p\mu^{+}\mu^{-}$ decay111The
inclusion of charge conjugated processes is implied throughout this note. is a
flavour changing neutral current process, allowed only at loop level in the
standard model (SM). The process is dominated by long-distance contributions
for a predicted branching fraction of $\mathcal{B}(\mathchar
28934\relax^{+}\rightarrow p\mu^{+}\mu^{-})\in[1.6,9.0]\times 10^{-8}$ [1],
while the short-distance SM contributions are suppressed at a branching
fraction of about $10^{-12}$. Evidence for this decay was seen by the HyperCP
experiment [2] with a measured branching fraction $\mathcal{B}(\mathchar
28934\relax^{+}\rightarrow p\mu^{+}\mu^{-})=(8.6^{+6.6}_{-5.4}\pm 5.5)\times
10^{-8}$, which is compatible with the SM. Remarkably, the three observed
decays have almost the same dimuon pair invariant mass of $m_{X^{0}}=214.3\pm
0.5{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, which lies close to the
kinematic limit. Such a distribution, if confirmed, would point towards a
decay with an intermediate particle coming from the $\mathchar
28934\relax^{+}$ baryon and decaying into two muons, _i.e._ a $\mathchar
28934\relax^{+}\rightarrow pX^{0}(\rightarrow\mu^{+}\mu^{-})$ decay, which
would constitute evidence of physics beyond the SM (BSM). Various BSM
explanations have been proposed to explain the HyperCP result. The
intermediate $X^{0}$ particle could be for example a light pseudoscalar Higgs
boson [3, 4] in a NMSSM or 2HDM model, or a sgoldstino [5, 6] in various
supersymmetric models. Other interpretations and implications can be found in
Ref. [7, 8, 9, 10, 11, 12, 13]; in general a pseudoscalar particle is favoured
over a scalar one and a lifetime of order $10^{-14}\mathrm{{\,s}}$ is
estimated for the former case. This lifetime would correspond to a prompt
$X^{0}$ signal, decaying in the same vertex as the $\mathchar 28934\relax^{+}$
baryon, in any present search for this particle. Attempts to confirm the
presence of this $X^{0}$ particle have been made by different experiments in
various initial and final states without finding any signal [14, 15, 16, 17,
18, 19, 20, 21], and in LHCb through the final states
$B^{0}_{(s)}\rightarrow\mu^{+}\mu^{-}\mu^{+}\mu^{-}$ [22] and
$B^{0}\rightarrow K^{\ast 0}\mu^{+}\mu^{-}$ [23]. However, the search for the
$\mathchar 28934\relax^{+}\rightarrow p\mu^{+}\mu^{-}$ decay has not been
repeated, mainly due to the absence of experiments with large hyperon
production and due to experimental difficulties.

Hyperons are produced copiously in high energy proton-proton collisions at the
Large Hadron Collider (LHC). A search for $\mathchar
28934\relax^{+}\rightarrow p\mu^{+}\mu^{-}$ decays at the LHCb experiment
could therefore confirm or disprove the HyperCP evidence, and measure its
branching fraction. This search was also suggested in Ref. [24]. In this
report, the search for the $\mathchar 28934\relax^{+}\rightarrow
p\mu^{+}\mu^{-}$ decay is presented as performed using $pp$ collision data
recorded by the LHCb experiment at centre-of-mass energies $\sqrt{s}=7$ and
$8\mathrm{\,Te\kern-1.00006ptV}$, corresponding to an integrated luminosity of
$3\mbox{\,fb}^{-1}$. The measurement described here is detailed in Ref. [25].

##  2 Detector and data samples

The LHCb detector is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks, and is detailed elsewhere [26, 27]. The online
event selection is performed by a trigger, which consists of a hardware stage,
based on information from the calorimeter and muon systems, followed by two
software stages, the first performing a preliminary event reconstruction based
on partial information while the second applying a full event reconstruction.
Each of the three trigger stages is divided into several parallel trigger
lines dedicated to different kinds of signals. Decay particles involved in
this analysis are often not able to activate one or more trigger stages, owing
mainly to their soft transverse momenta. Nevertheless, since $\mathchar
28934\relax^{+}$ baryons are copiously produced in the $pp$ collisions
recorded by LHCb, the present search can be performed on the events already
recorded. In the offline processing, trigger decisions are associated with
reconstructed candidates. A trigger decision on a particular line or on the
full trigger can thus be ascribed to the reconstructed candidate, the rest of
the event or both; events triggered as such are defined respectively as
triggered on signal (TOS), independently of signal (TIS), or triggered on
both. The estimation of the trigger efficiency when no specific path is
selected is difficult, therefore a different strategy is adopted here: all the
candidates passing the selection are used in the search for $\mathchar
28934\relax^{+}\rightarrow p\mu^{+}\mu^{-}$ decays, while only the TIS
candidates are used to convert the event yield into a branching fraction. This
ensures a partial cancellation of the TIS trigger efficiency between signal
and normalisation channels. Furthermore, control channels with large
statistics can be exploited to estimate the trigger efficiency directly on
data by measuring the overlap of events which are TIS and TOS simultaneously
[28].

##  3 Analysis strategy and selection

After the trigger, a loose selection is applied based on geometric and
kinematic variables. Afterwards candidates are selected by means of a
multivariate selection based on a boosted decision trees algorithm (BDT) [29,
30]. The final search datasets are obtained by rejecting the background with a
cut on the BDT output and on particle identification variables. The signal
yield is obtained from a fit to the $p\mu^{+}\mu^{-}$ invariant mass and is
converted into a branching fraction by normalising to the $\mathchar
28934\relax^{+}\rightarrow p\pi^{0}$ control channel. To avoid experimenter
bias, candidates in the signal regions were not examined until the analysis
procedure had been finalised.

The analysis is designed in order to search for possible peaks in the dimuon
invariant mass, pointing towards unknown intermediate particles. The
resolution on the dimuon invariant mass as a function of the mass itself is
shown in Fig. 1(a). The selection is devised such that no fake structures are
induced in the dimuon invariant mass distribution of signal candidates. The
signal efficiency varies as a function of the dimuon invariant mass and is
shown in Fig. 1(b). The efficiency is larger at lower dimuon mass owing to the
larger recoil momentum against the proton, which gives larger minimum
transverse momentum to the two muons, ensuring a better tracking efficiency.

\begin{overpic}[width=212.47617pt]{Fig2a.pdf} \put(20.0,15.0){\small(a)}
\end{overpic}\begin{overpic}[width=212.47617pt]{Fig2b.pdf}
\put(20.0,15.0){\small(b)} \end{overpic}

Figure 1: Resolution (a) on the dimuon invariant mass $m_{\mu^{+}\mu^{-}}$ as
a function of the mass itself for simulated $\mathchar
28934\relax^{+}\rightarrow p\mu^{+}\mu^{-}$ decays in LHCb. Efficiency (b) for
simulated $\mathchar 28934\relax^{+}\rightarrow p\mu^{+}\mu^{-}$ decays in
LHCb as a function of the dimuon invariant mass.

Together with the signal $\mathchar 28934\relax^{+}\rightarrow
p\mu^{+}\mu^{-}$ decay, the following channels are also selected in data: the
$\mathchar 28934\relax^{+}\rightarrow p\pi^{0}$ decay as normalisation
channel, the $\mathchar
28934\relax^{+}\rightarrow{\overline{p}}\mu^{+}\mu^{+}$ candidate decay as
control channel for the combinatorial background, and the
$K^{+}\rightarrow\pi^{+}\pi^{-}\pi^{+}$ decay as a control channel for
different parts of the efficiency evaluation. Candidate $\mathchar
28934\relax^{+}\rightarrow p\mu^{+}\mu^{-}$ decays are selected by combining
two good quality oppositely charged tracks with muon identification with a
third track with proton identification. The three tracks are required to form
a good quality secondary vertex (SV), displaced from any $pp$ interaction
vertex (PV) requiring a measured $\mathchar 28934\relax^{+}$ lifetime greater
than 6 ps. Only $\mathchar 28934\relax^{+}$ candidates with transverse
momentum $\mbox{$p_{\mathrm{T}}$}>0.5{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$,
$\chi^{2}_{\rm IP}<36$ and cosine of the angle between the flight direction
and the reconstructed momentum larger than 0.9 are retained. Candidate
$\mathchar 28934\relax^{+}\rightarrow p\mu^{+}\mu^{-}$ decays are considered
only if the invariant mass satisfies $|m_{p\mu^{+}\mu^{-}}-m_{\mathchar
28934\relax^{+}}|<500{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, where
$m_{\mathchar 28934\relax^{+}}$ is the known mass of the $\mathchar
28934\relax^{+}$ particle [31]. A large background component is present in
data due to $\mathchar 28931\relax^{0}\rightarrow p\pi^{-}$ decays, where the
pion is misidentified as a muon and the two tracks are combined with a third
one. This is vetoed discarding candidates having a $p\mu^{-}$ pair mass within
5 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ from the $\mathchar
28931\relax^{0}$ known mass when calculated with the $p\pi^{-}$ mass
hypothesis. Possible backgrounds from exclusive decays peaking in the
$p\mu^{+}\mu^{-}$ invariant mass have been examined, including
$K^{+}\rightarrow\pi^{+}\pi^{-}\pi^{+}$ and
$K^{+}\rightarrow\pi^{+}\mu^{-}\mu^{+}$ decays and various hyperon decays, and
none has been found to contribute. In case of multiple candidates in a single
event, all are retained in the selection. Multiple candidates are present in
5% of the events after the initial selection, while no multiple candidate is
present in the final selection. The selection for the control channel
$\mathchar 28934\relax^{+}\rightarrow{\overline{p}}\mu^{+}\mu^{+}$ is
identical to that of the signal but considering same-sign dimuon pairs.
Candidate $\mathchar 28934\relax^{+}\rightarrow p\pi^{0}$ decays are selected
by combining one good quality track with proton identification with a
$\pi^{0}$ reconstructed in the $\pi^{0}\rightarrow\gamma\gamma$ mode from two
clusters in the calorimeter. The selection of this decay is similar to that of
the signal, but places a tighter requirement on the proton identification, and
on the transverse momenta of the daughters in order to reduce the high
combinatorial background. Finally, candidate
$K^{+}\rightarrow\pi^{+}\pi^{-}\pi^{+}$ decays are selected from three good
quality tracks, one of which is oppositely charged with respect to the other
two; the selection is similar but tighter than that of the signal to cope with
the high level of combinatorial background. See Ref. [25] for further details
on the selection criteria of the different channels.

The sample of $\mathchar 28934\relax^{+}\rightarrow p\mu^{+}\mu^{-}$
candidates in data after the selection is dominated by combinatorial
background, part of which is due to misidentified particles. This is rejected
by cuts on the BDT and on multivariate particle identification variables [27]
on the muons and on the proton. The BDT variable combines geometric and
kinematic variables chosen so that the dependence on the $p\mu^{+}\mu^{-}$
invariant mass and on the dimuon invariant mass is linear and small to avoid
biases. The BDT is optimised using simulated samples of $\mathchar
28934\relax^{+}\rightarrow p\mu^{+}\mu^{-}$ events for the signal and
candidates from the sidebands of the $\mathchar
28934\relax^{+}\rightarrow{\overline{p}}\mu^{+}\mu^{+}$ selection in data for
the background. The final cut values were chosen in order to optimise the
sensitivity to a signal evidence for the smallest possible branching fraction
[32]. No BDT selection is applied to the normalisation and control channels.

##  4 Normalisation

The number of signal candidates in the TIS sample is converted into a
branching fraction with the formula

$\displaystyle\mathcal{B}(\mathchar 28934\relax^{+}\rightarrow
p\mu^{+}\mu^{-})=\frac{\varepsilon_{\mathchar 28934\relax^{+}\rightarrow
p\pi^{0}}}{\varepsilon_{\mathchar 28934\relax^{+}\rightarrow
p\mu^{+}\mu^{-}}}\frac{N_{\mathchar 28934\relax^{+}\rightarrow
p\mu^{+}\mu^{-}}}{N_{\mathchar 28934\relax^{+}\rightarrow
p\pi^{0}}}\mathcal{B}(\mathchar 28934\relax^{+}\rightarrow
p\pi^{0})=\alpha\cdot N_{\mathchar 28934\relax^{+}\rightarrow
p\mu^{+}\mu^{-}}\quad,$

where $\varepsilon$, $N$ and $\mathcal{B}$ are the efficiency, candidate yield
and branching fraction of the corresponding channel, respectively, and
$\alpha$ is the single event sensitivity.

The ratio of signal and normalisation channel efficiencies, which includes the
acceptance, the reconstruction efficiency of the final state particles and the
selection efficiency, is computed with samples of simulated events corrected
to take into account known differences between data and simulation. The
reconstruction efficiency for the $\pi^{0}$ is calibrated using the ratio of
$B^{+}\rightarrow J/\psi K^{\ast+}(\rightarrow K^{+}\pi^{0})$ and
$B^{+}\rightarrow J/\psi K^{+}$ decays reconstructed in data. The particle
identification efficiency of protons and muons is calibrated exploiting
control channels in data. Residual differences between data and simulation are
treated as sources of systematic uncertainty. The signal and normalisation
channels are required to be TIS at all trigger levels. The trigger efficiency
ratio is thus expected to be unity. However, small differences in the average
kinematic of the rest of the event in the two samples are present which cause
the ratio to be different. The ratio of trigger efficiencies is thus evaluated
with data-driven techniques [28] exploiting the large sample of
$K^{+}\rightarrow\pi^{+}\pi^{-}\pi^{+}$ decays. A systematic uncertainty is
assigned for the applicability of this method to the relatively soft events of
this analysis.

The invariant mass distribution of the $K^{+}\rightarrow\pi^{+}\pi^{-}\pi^{+}$
control channel candidates in data is shown in Fig. 2(a). A binned maximum
likelihood extended fit is performed to the invariant mass distribution. The
signal is described as an Hypatia function [33] while the background is
described by a second-order polynomial. A total of $(966\pm 2)\times 10^{3}$
$K^{+}\rightarrow\pi^{+}\pi^{-}\pi^{+}$ candidates is measured.

Figure 2: Invariant mass distribution of (a)
$K^{+}\rightarrow\pi^{+}\pi^{-}\pi^{+}$ candidates superimposed with the fit
to data; (b) fit to the distribution of the corrected mass $m_{\mathchar
28934\relax}^{\text{corr}}$ for $\mathchar 28934\relax^{+}\rightarrow
p\pi^{0}$ candidates superimposed with the fit to data.

The observed number of $\mathchar 28934\relax^{+}\rightarrow p\pi^{0}$
candidates is $(1\,711\pm 9)\times 10^{3}$ as obtained from a binned maximum
likelihood extended fit to the corrected invariant mass distribution. The
corrected invariant mass is defined as $m_{\mathchar
28934\relax}^{\text{corr}}=m_{p\pi^{0}}-m_{\pi^{0}}+m_{\pi^{0}}^{\text{PDG}}$
to correct for the $\pi^{0}$ mass reconstructed from the two photons. The
$\mathchar 28934\relax^{+}\rightarrow p\pi^{0}$ distribution is described as a
Gaussian function with a power tail on the right side, while the background is
described by a modified ARGUS function [34]. The invariant mass distribution
is shown in Fig. 2(b), superimposed with the fit. The single event sensitivity
is $\alpha=(1.1\pm 0.6)\times 10^{-8}$, where the uncertainty is dominated by
the aforementioned systematic uncertainties, and corresponding to about
$4\times 10^{11}$ $\mathchar 28934\relax^{+}$ particles produced in the LHCb
acceptance in the full dataset in TIS events. This corresponds to $4.6\pm 4.2$
$\mathchar 28934\relax^{+}\rightarrow p\mu^{+}\mu^{-}$ candidates assuming a
branching fraction of $(5\pm 4)\times 10^{-8}$, to cover the SM predicted
range.

The observed number of signal $\mathchar 28934\relax^{+}\rightarrow
p\mu^{+}\mu^{-}$ events is obtained with a fit to the $p\mu^{+}\mu^{-}$
invariant mass distribution in the range
$1149.6<m_{p\mu^{+}\mu^{-}}<1409.6{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
The signal distribution is described by an Hypatia function [33]. The mass
resolution and scale are calibrated using the control channel
$K^{+}\rightarrow\pi^{+}\pi^{-}\pi^{+}$ and comparing data and simulation
distributions. No bias is seen in the peak position, while a 25% correction to
the resolution has to be applied to match the one observed in data. A
resolution of $4.28\pm 0.19{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ is used
as width of the signal $\mathchar 28934\relax^{+}\rightarrow p\mu^{+}\mu^{-}$
distribution. The resolution is allowed to vary in the fit but constrained to
the central value with a Gaussian constraint. The combinatorial background is
described as a modified ARGUS function, with all parameters left free with
exception of the threshold which is fixed to the kinematic limit.

While the normalisation is available for the TIS sample, further studies are
needed to calculate the trigger efficiency for the full sample. Since the TIS
dataset is a sub-sample of the full sample, the single event sensitivity of
the latter will be equal or better than the one of the former.

##  5 Results and discussion

The invariant mass distribution of the $\mathchar 28934\relax^{+}\rightarrow
p\mu^{+}\mu^{-}$ candidates in data is shown in Fig. 3(a) and (b) for the full
and TIS datasets, respectively. A significant signal is present in the full
dataset. The significance of the signal is of $4.0\,\sigma$, obtained from the
comparison of the likelihood value of the full fit with that of the
background-only fit, and includes the relevant systematic uncertainties as
gaussian constrains to the likelihood. A total of $12.9^{+5.1}_{-4.2}$ signal
candidates is observed. The signal in the TIS sample is found not to be
significant. This is due to the observed signal candidates being needed for
the related event to switch on at least one of the LHCb trigger stages. From
the absence of a significant signal in the TIS dataset an upper limit on the
$\mathchar 28934\relax^{+}\rightarrow p\mu^{+}\mu^{-}$ branching fraction is
derived, using the CLs method [35], of $\mathcal{B}(\mathchar
28934\relax^{+}\rightarrow p\mu^{+}\mu^{-})<6.3\times 10^{-8}$ at 95%
confidence level (CL).

\begin{overpic}[width=216.81pt]{Fig6a.pdf} \put(18.0,57.0){\small(a)}
\end{overpic}\begin{overpic}[width=216.81pt]{Fig6b.pdf}
\put(18.0,57.0){\small(b)} \end{overpic}

Figure 3: Invariant mass distribution for $\mathchar
28934\relax^{+}\rightarrow p\mu^{+}\mu^{-}$ candidates: (a) all the candidates
and (b) only the candidates satisfying the TIS requirement.

Considering candidates in the full selection, a scan for possible signals in
the dimuon invariant mass is performed, restricted to within two times the
resolution in the $\mathchar 28934\relax^{+}\rightarrow p\mu^{+}\mu^{-}$
invariant mass around the known $\mathchar 28934\relax^{+}$ mass. The scan is
performed considering a single Gaussian function for a putative $X^{0}$
resonance (signal) and a linear distribution for the remaining candidates
(background). In Fig. 4(a) the local p-value of the background-only hypothesis
as a function of the dimuon mass is shown. No significant signal is found. In
Fig. 4(b) the fit corresponding to a mass of
$214.3{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ is shown. The fitted number of
events for this hypothesis is $1.6\pm 1.9$, corresponding to a fraction
$0.078\pm 0.092$ of the considered candidates.

\begin{overpic}[width=212.47617pt]{Fig7a.pdf} \put(70.0,45.0){\small(a)}
\end{overpic}\begin{overpic}[width=212.47617pt]{Fig7b.pdf}
\put(50.0,45.0){\small(b)} \end{overpic}

Figure 4: Plot of the local p-value as a function of the dimuon mass as
obtained from the fits (see text) in (a) and (b) fit for a signal with mass
$m_{X^{0}}=214.3{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.

In summary, a search for the $\mathchar 28934\relax^{+}\rightarrow
p\mu^{+}\mu^{-}$ rare decay is performed by the LHCb experiment at centre-of-
mass energies $\sqrt{s}=7$ and $8\mathrm{\,Te\kern-1.00006ptV}$, corresponding
to an integrated luminosity of $3\mbox{\,fb}^{-1}$. Evidence for the
$\mathchar 28934\relax^{+}\rightarrow p\mu^{+}\mu^{-}$ decay is found in the
full dataset, albeit not found when requiring the events to be triggered
independently of the signal decay. The observed signal candidates show a
dimuon invariant mass distribution consistent with phase space; no significant
peak consistent with an intermediate particle is found in the dimuon invariant
mass distribution. The upper limit on the branching fraction of the $\mathchar
28934\relax^{+}\rightarrow p\mu^{+}\mu^{-}$ decay is $6.3\times 10^{-8}$ at
95% CL for a SM-like signal.

## References

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