# A dynamic approach for the zeros of the Riemann zeta function - Collision
and repulsion

Yochay Jerby  Yochay Jerby, Faculty of Sciences, Holon Institute of
Technology, Holon, 5810201, Israel<EMAIL_ADDRESS>
###### Abstract.

For $N\in\mathbb{N}$ consider the $N$-th section of the approximate functional
equation

$\zeta_{N}(s)=\sum_{n=1}^{N}B_{n}(s),$

where

$B_{n}(s)=\frac{1}{2}\left[n^{-s}+\chi(s)\cdot n^{s-1}\right].$

Our aim in this work is to introduce a new approach for the Riemann hypothesis
by studying the way pairs of consecutive zeros of $\zeta_{N}(s)$ change with
respect to $N$.

For the initial stage, it is known that the non-trivial zeros of
$\zeta_{1}(s)$ all lie on the critical line $Re(s)=\frac{1}{2}$. In the region
$2N\leq Im(s)\leq 2\pi(N+1)$ the function $\zeta_{N}(s)$ serves as an
approximation of $\zeta(s)$ itself, and it was conjectured by Spira that in
this region $\zeta_{N}(s)$ also admits zeros only on the critical line.

We show that the appearance of zeros of a section off the critical line can be
realized as the result of two consecutive zeros meeting and pushing each other
off the critical line as $N$ changes, a process to which we refer to as _a
collision of zeros_. Based on a study of the properties of $\zeta_{N}(s)$, we
suggest a way of re-arranging the order of summation of the elements
$B_{n}(s)$ in $\zeta_{\left[\frac{Im(s)}{2}\right]}(s)$ that is expected to
avoid collisions altogether, we refer to such a re-arrangement as a _repelling
re-arrangement_. In particular, establishing that the repelling re-arrangement
indeed avoids collisions for _any_ pair of zeros would imply RH.

##  1\. Introduction and Summary of Main Results

Let $\zeta(s)$ be the Riemann zeta function for $s=\sigma+it$. In the range
$1<\sigma$ the function is defined as

$\zeta(s):=\sum_{n=1}^{\infty}\frac{1}{n^{s}},$ (1)

and is extended analytically to be defined on the whole complex plane. The
Riemann Hypothesis postulates:

The Riemann Hypothesis: The non-trivial zeros $\rho$ of the Riemann zeta
function, $\zeta(s)$, all lie on the critical line $\sigma=\frac{1}{2}$.

In fact, from the direct definition of $\zeta(s)$ itself, as given for
instance in Eq. 1, it is rather hard to decipher any useful insight regarding
the zeros $\rho$ and their location. In practice (beginning from Riemann
himself), values of $\zeta(s)$ are not computed via the direct definition, but
typically via some form of an approximate functional equation, see [11, 12,
13, 21]. In the 1960’s Robert Spira conducted a theoretical and numerical
study of the zeros of partial sums (sections) of the classical approximate
functional equation, see [23, 24]. That is, Spira considered the zeros of the
$N$-th sections of the approximate functional equation

$\zeta_{N}(s)=\sum_{n=1}^{N}B_{n}(s),$ (2)

where

$B_{n}(s)=\frac{1}{2}\left[n^{-s}+\chi(s)\cdot n^{s-1}\right],$ (3)

and

$\chi(s):=2^{s}\pi^{s-1}sin\left(\frac{\pi s}{2}\right)\Gamma(1-s)$ (4)

is the function appearing in the functional equation

$\zeta(s)=\chi(s)\zeta(1-s).$ (5)

For $N=1$ and $N=2$ (for $t$ sufficiently large) Spira showed that
$\zeta_{N}(s)$ satisfy the Riemann hypothesis and admit zeros only on the
critical line $\sigma=\frac{1}{2}$. Due to the functional equation, the
function $\zeta_{N}(s)$ serves as an approximation of $\zeta(s)$ in the region
$\sqrt{2\pi N}\leq t\leq 2\pi N$. In fact, Spira conjectures the following RH
for sections:

Conjecture (Spira [23]): All the zeros of the section $\zeta_{N}(s)$ in the
region $\sqrt{2\pi N}\leq t\leq 2\pi N$ lie on the critical line
$\sigma=\frac{1}{2}$.

In his studies it is apparent that Spira mainly considered the properties of
zeros of $\zeta_{N}(s)$ separately, for given $N$ at a time. Our aim in this
work is to present a new approach to Spira’s conjecture. The main feature is
that rather than studying the zeros of a given section $\zeta_{N}(s)$
independently, we are interested in studying the way the zeros of the sections
$\zeta_{N}(s)$ change with respect to $N$. Concretely, let us summarize the
main points of our approach:

  1. (1)

Rouche’s theorem implies the existence of a one-to-one correspondence between
the zeros of $\zeta_{N}(s)$ and $\zeta_{N+1}(s)$ in the critical strip. In
particular, no ”new” zeros are created in the critical strip during the
transition from one section to the other. We are thus interested in the way in
which the zeros of the sections $\zeta_{N}(s)$ change as $N$ changes from the
first section $\zeta_{1}(s)$, whose zeros are known to lie on the critical
line, to the $\left[\frac{t}{2}\right]$-th section, whose zeros are expected
to lie on the critical line by Spira’s conjecture.

  2. (2)

Even-though the zeros are known to begin at the initial stage of $N=1$ on the
critical line, and are expected to eventually also lie on the critical line
for $N=\left[\frac{t}{2}\right]$, some of the intermediary sections
$\zeta_{N}(s)$ might violate RH. That is their zeros do not nessecerally need
to lie on the critical line for any $1\leq N\leq\left[\frac{t}{2}\right]$. We
observe, however, that zeros of $\zeta_{N}(s)$ can appear off the critical
line only if a process to which we refer as _collision_ occurred between a
pair of consecutive zeros in a previous stage. This leads us to study the
interactions between pairs of zeros as $N$ changes.

  3. (3)

The question is thus, could collisions between a given pair of zeros be
avoided? By definition, the section $\zeta_{\left[\frac{t}{2}\right]}(s)$ of
interest is given as the sum of the elements

$B_{n}(s)=\frac{1}{2}\left[n^{-s}+\chi(s)\cdot n^{s-1}\right]$ (6)

for the first $n=1,...,\left[\frac{t}{2}\right]$. The initial approach
described above is based on summing the elements $B_{n}(s)$ in consecutive
order (giving rise to the sections $\zeta_{N}(s)$) until the
$\left[\frac{t}{2}\right]$-th element is added. However, the problem with this
naive approach is that the unwanted collisions could occur between various
pairs of zeros, pushing those zeros away from the critical line at certain
stages.

The question can thus be rephrased as follows: can the summation order of the
elements $B_{n}(s)$ be re-arranged in a different manner so that collisions
would be avoided altogether? That is, is it possible that the collisions are
not an essential phenomena but rather a by-product of the specific consecutive
order of summation considered?

We give various theoretical justifications suggesting that the answer to this
question is affirmative. In fact, based on a few fundamental observations
regarding the way the sections $\zeta_{N}(s)$ change with respect to $N$, we
suggest a well-defined re-arrangement of the order of summation. We conjecture
that this re-arrangement avoids collisions for any pair of consecutive zeros
altogether, and hence refer to it as a _repelling re-arrangement_.

In short our our observations could be summarized as follows:

  * •

The non-trivial zeros of $\zeta(s)$ are ”born” as zeros of $\zeta_{1}(s)$ on
the critical line, which are regulated and well-understood, and dynamically
develop by gradually adding the first $\left[\frac{t}{2}\right]$ elements of
$B_{n}(s)$ to $\zeta_{1}(s)$ to obtain $\zeta_{\left[\frac{t}{2}\right]}(s)$.

  * •

We conjecture that, when the addition is done via the repelling re-
arrangement, no collision occurs between consecutive zeros and, hence, the
zeros always remain on the critical line. This includes the final stage, where
their position _on the critical line_ is identical to that of the zeros of
$\zeta(s)$, up to a negligible error. In particular, the non-trivial zeros of
$\zeta(s)$ must lie on the critical line, that is, they satisfy RH.

Let us note that throughout the work we would also be concerned with a variant
of the classical sections $\zeta_{N}(s)$ to which we refer as the accelerated
sections $\widetilde{\zeta}_{N}(s)$. The accelerated sections are given as
partial sums of the Euler transformation of series of the defining sum of
$\zeta(s)$ given by Eq. 1. Everything mentioned for the classical sections
applies to the accelerated sections as well. However, the advantage of the
accelerated sections $\widetilde{\zeta}_{N}(s)$ is twofold. First, the
approximation of zeta afforded by them is far superior to that given by the
classical sections $\zeta_{N}(s)$. Moreover, their change with respect to $N$
is ”smoothened” relative to that of the classical sections.

The rest of this work is devoted to explaining and expanding in detail on
Points (1)-(3) and is organized as follows: In Section 2 we introduce the
Euler transformation of series procedure for $\zeta(s)$ and the corresponding
accelerated sections $\widetilde{\zeta}_{N}(s)$. An initial discussion on the
analytical distinctions between the classical and accelerated sections is
presented. In Section 3 we discuss the zeros of $\zeta_{1}(s)$, review their
recent representation in terms of the Lambert function due to Franca-LeClair
given in [6, 7] as well as present an alternative new description. In Section
4 we study the zeros of the sections $\zeta_{N}(s)$ and give examples for
their collisions as $N$ changes from zero to $\left[\frac{t}{2}\right]$.
Furthermore, new interpretations of Gram’s law are presented, see Remark 3.2.
In Section 5, based on observations from Section 2 and Section 4, we introduce
the repelling re-arrangement and illustrate how it leads to avoiding of
collisions. We consider the Davenport-Heilbronn function $\mathcal{D}(s)$, and
show how the various phenomena discussed for zeta are violated for
$\mathcal{D}(s)$, see Remark 5.3. Relations to the Montgomery pair correlation
conjecture is also considered, see Remark 5.4. Finally, in Section 6 we
present a summary and concluding remarks.

##  2\. The Euler transformation of series for $\zeta(s)$ and its comparison
to the classical sum

For $1<\sigma$ the Riemann zeta is defined by the series

$\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}.$ (7)

Although that for $\sigma\leq 1$ the series is not converging it is
nevertheless interesting to consider how it behaves in this region. Consider
the partial sums

$S_{N}(s):=\sum_{n=1}^{N}\frac{1}{n^{s}}$ (8)

for $N\in\mathbb{N}$. The following Fig. 1 shows the values of $ln\lvert
S_{N}(\frac{1}{2}+17500i)\rvert$ for $N=1,...,5000$ (blue) together with the
value $ln\lvert\zeta(\frac{1}{2}+17500i)\rvert$ (orange):

Figure 1. The values of $ln\lvert S_{N}(\frac{1}{2}+17500i)\rvert$ for
$N=1,...,5000$ (blue) together with the value
$ln\lvert\zeta(\frac{1}{2}+17500i)\rvert$ (orange).

Figure 1 could be considered as illustrating the typical behaviour of the
partial sums $S_{N}(s)$. In particular, one can see from Fig. 1 two features:

  1. (1)

Although the series is non-convergent, for $N>>0$ big enough111It should be
noted that if $N$ is taken too big $\zeta_{N}(s)$ what start deviating from
the value of $\zeta(s)$. the partial sums $S_{N}(s)$ do eventually serve as
approximations of $\zeta(s)$.

  2. (2)

The partial sums are $S_{N}(s)$ seen to fluctuate around other values before
stabilizing around the final value of $\zeta(s)$ for $N$ big enough.

In fact, the phenomena presented in Fig. 1 is explained by the classical
approximate functional equation of Hardy and Littlewood. The classical
approximate functional equation for the Riemann zeta function was proven by
Hardy and Littlewood in the series of works [11, 12, 13]. For $0\leq\sigma\leq
1$, the theorem states that the following holds

$\zeta(s)=\sum_{n\leq x}n^{-s}+\chi(s)\left(\sum_{n\leq
y}n^{s-1}\right)+O\left(x^{-\sigma}+x^{\frac{1}{2}-\sigma}y^{-\frac{1}{2}}\right),$
(9)

when $x,y\geq 1$ are such that $|t|=2\pi xy$. In particular, the AFE explains
what are the sub-values around which $S_{N}(s)$ fluctuates before stabilizing
around $\zeta(s)$. These are exactly the values
$\zeta(s)-\chi(s)\left(\sum_{n=1}^{M}n^{s-1}\right)$ and the region of values
of $N$ for which these fluctuations occurs is roughly the region

$\left[\frac{t}{2(M+1)\pi}\right]\leq N\leq\left[\frac{t}{2M\pi}\right].$ (10)

This rephrasing of the AFE in view of the behaviour expressed in Fig. 1 would
be of importance in the discussion of the following Section 6.

Let us now turn to consider the Euler transformation of series for
$\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}$. Recall that to a given
alternating series (convergent or divergent)

$S=\sum_{n=1}^{\infty}(-1)^{n-1}a_{n}$ (11)

one can apply the highly classical procedure of Euler’s transformation of
series, see [5, 10, 17]. Concretely, one can re-write the series as

$S=\sum_{n=0}^{\infty}\frac{\Delta^{n}a_{1}}{2^{n+1}},$ (12)

where

$\Delta^{n}a_{1}:=\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}a_{k+1}.$ (13)

As a direct application of the transformation to the defining series of zeta,
$\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}$, itself, that is to the sequence

$a_{n}(s):=\frac{(-1)^{n-1}}{n^{s}},$ (14)

we obtain the following global formula222Formula Eq. 15 is a variant of the
Hasse-Sondow global formula, obtained by applying Euler acceleration to the
defining series of the Dirichlet eta function $\eta(s)$ instead of Eq. 1, see
[9, 22].:

######  Proposition 2.1.

The following formula

$\zeta(s)=\sum_{n=0}^{\infty}\widetilde{A}(s,n),$ (15)

where

$\widetilde{A}(s,n):=\frac{1}{2^{n+1}}\sum_{k=0}^{n}\binom{n}{k}\frac{1}{(k+1)^{s}},$
(16)

holds for any $s\in\mathbb{C}$.

Let us compare the behaviour of the transformed series to that of the
classical one. Denote the partial sums of the transformed series by

$\widetilde{S}_{N}(s)=\sum_{n=0}^{N}\widetilde{A}(s,n).$ (17)

for $N\in\mathbb{N}$. Consider the following Fig. 2:

Figure 2. (a) Values of $ln\lvert\widetilde{S}_{N}(\frac{1}{2}+1200i)\rvert$
for $N=1,...,600$ (blue) together with the value
$ln\lvert\zeta(\frac{1}{2}+1200i)\rvert$ (orange) (b) Values of $ln\lvert
S_{N}(\frac{1}{2}+1200i)\rvert$ for $N=1,...,600$ (blue) together with the
value $ln\lvert\zeta(\frac{1}{2}+1200i)\rvert$ (orange).

In Fig. 2 one sees the following two features showing the advantage of the
accelerated series over the classical one:

  1. (1)

Contrary to the classical case, the transformed partial sums
$\widetilde{S}_{N}(s)$ seem to be a superb approximation of $\zeta(s)$ for
$N>>0$ big enough. In fact, in the range $2N\leq t\leq 2(N+1)$ one has the
following approximation with exponentially decaying error term

$\zeta(s)=\widetilde{S}_{N}(s)+O(e^{-\omega\lvert t\rvert})$ (18)

for $\omega>0$ is a certain positive constant, which is a variant of our
recent result in the setting of the Hasse-Sondow formula, see [16]. This
approximation, whose error term is of exponentially decaying error, is far
superior to the approximation afforded by the classical approximate functional
equation, whose error term is only algebraic.

  2. (2)

Although the application of the transformation seems to _smoothen_ the
behaviour of $\widetilde{S}_{N}(s)$ with respect to $N$, it still bares much
overall similarity to the behaviour of the original classical sums $S_{N}(s)$.
In fact, this smoothing feature could be explained by noting that by changing
the order of summation one can also write

$\overline{S}_{N}(s)=\sum_{k=0}^{N}\widetilde{a}(k,N)(k+1)^{-s}.$ (19)

where the constants are given by

$\widetilde{a}(k,N):=\sum_{n=k}^{N}\frac{1}{2^{n+1}}\binom{n}{k}.$ (20)

In comparison, the classical sections could be written via a similar formula
with $a(k,N)=1$. In other words, the transformed sections
$\widetilde{S}_{N}(s)$ could be considered as adding the weights
$\widetilde{a}(k,N)$ instead of the trivial weights for the classical
sections.

In particular, let us define the accelerated $N$-th sections of the global
representation

$\widetilde{\zeta}_{N}(s)=\sum_{n=0}^{N}\widetilde{B}_{n}(s),$ (21)

where

$\widetilde{B}_{n}(s)=\frac{1}{2}\left[\widetilde{A}(s,n)+\chi(s)\cdot\widetilde{A}(1-s,n)\right].$
(22)

In what follows we would typically compare the properties of the classical
sections $\zeta_{N}(s)$ to those of the accelerated ones
$\widetilde{\zeta}_{N}(s)$.

##  3\. On the zeros of $\zeta_{1}(s)$ and their representations

In this section we consider the zeros of the first section
$\zeta_{1}(s)=2\widetilde{\zeta}_{0}(s)=1+\chi(s)$. The first part of this
section contains review of known results of Spira and Franca-LeClair, see [6,
23, 24]. The zeros of $\zeta_{1}(s)$ were studied in the 1960’s by Spira who
showed that all the zeros in the critical strip must lie on the critical line.
Moreover, Spira also observed that between any two Gram points333It should be
noted that Spira conducted his studies before the establishment of the current
notations of the Lambert function. lies a zeros of $\zeta_{1}(s)$, and vice
versa, a feature to which we shall return shortly.

Consider the following Fig. 3 which shows
$ln\lvert\zeta\left(\frac{1}{2}+it\right)\rvert$ (blue) and
$ln\lvert\zeta_{1}\left(\frac{1}{2}+it\right)\rvert$ (orange) for $0\leq t\leq
50$:

Figure 3. Graphs of $ln\lvert\zeta\left(\frac{1}{2}+it\right)\rvert$ (blue)
and $ln\lvert\zeta_{1}\left(\frac{1}{2}+it\right)\rvert$ (orange) for $0\leq
t\leq 50$.

As one can see, the zeros of $\zeta_{1}(s)=1+\chi(s)$ already serve as an
initial crude approximation of the zeros of $\zeta(s)$ on the critical line,
in the prescribed region. Recently, the zeros of $1+\chi(s)$ were further
studied (to a certain extent re-discovered) by Franca and LeClair in [6, 7],
where they showed that they can be described in terms of the Lambert function,
see [1, 14].

For large $t$ we have by Stirling’s formula

$\Gamma(\sigma+it)=\sqrt{\frac{2\pi}{\sigma+it}}\left(\frac{\sigma+it}{e}\right)^{\sigma+it}\left(1+O\left(\frac{1}{t}\right)\right).$
(23)

It also follows that

$\chi(\sigma+it)=\left(\frac{2\pi}{t}\right)^{\sigma+it-\frac{1}{2}}e^{i(t+\frac{\pi}{4})}\left(1+O\left(\frac{1}{t}\right)\right).$
(24)

Hence, define

$\overline{\chi}(\sigma,t):=\left(\frac{2\pi}{t}\right)^{\sigma+it-\frac{1}{2}}e^{i(t+\frac{\pi}{4})}.$
(25)

Consider the equation

$1+\overline{\chi}(\sigma,t)=1+\left(\frac{2\pi}{t}\right)^{\sigma+it-\frac{1}{2}}e^{i(t+\frac{\pi}{4})}=0.$
(26)

Taking absolute value implies $\sigma=\frac{1}{2}$ while taking argument
implies the equation

$\frac{t}{2\pi}\cdot ln\left(\frac{t}{2\pi e}\right)=n-\frac{11}{8}.$ (27)

Recall that the Lambert function is a multivalued function given by the
branches of the inverse function of $we^{w}$. For each $r$ there is one
branch, denoted $W_{r}(z)$ such that

$W_{r}(z)e^{W_{r}(z)}=z.$ (28)

Over the real numbers only the two branches $W_{0}(x)$ (also called the
principal branch) and $W_{-1}(x)$ are required. From Eq. 35 Franca and LeClair
deduced that the zeros of $1+\chi(s)$ can be approximated by
$\rho^{0}_{n}=\frac{1}{2}+it^{0}_{n}$ where $t^{0}_{n}$ are the solutions of

$t^{0}_{n}=\frac{(8n-11)\pi}{4W_{0}(\frac{8n-11}{8e})}.$ (29)

More generally, let us consider the functions

$B_{k}(s):=k^{-s}+\chi(s)\cdot k^{s-1}.$

By a similar argument the solutions of the equation $B_{k}(s)=0$ lie on the
critical line and their imaginary parts are given by the following equation

$\frac{t}{2\pi}\cdot ln\left(\frac{t}{2\pi k^{2}e}\right)=n-\frac{11}{8}.$

Let us define

$\begin{array}[]{ccc}\widetilde{t}^{+}_{k}(m)=\frac{(8m-3)\pi}{4W_{0}\left(\frac{8m-3}{8k^{2}e}\right)}&;&\widetilde{t}^{-}_{k}(m)=\frac{(5-8m)\pi}{4W_{-1}\left(\frac{5-8m}{8k^{2}e}\right)}.\end{array}$

We have the following description of the zeros of $B_{k}(s)$ on the critical
line, generalizing the Franca-LeClair formula for the case $k=1$:

######  Proposition 3.1.

Let $s^{k}_{m}=\frac{1}{2}+it^{k}_{m}$ be the zeros of $B_{k}(s)$ on the
critical line. Then $t^{k}_{m}=-\overline{t}^{k}_{-m}$ and for the zeros in
the upper half plane with $t^{k}_{m}>0$ the following holds:

  1. (1)

$t^{k}_{m}=\widetilde{t}^{-}_{k}(m)$ for $1\leq m\leq k^{2}$.

  2. (2)

$t^{k}_{m}=\widetilde{t}^{+}_{k}(m-2k^{2})$ for $k^{2}+1\leq m$.

As an illustration, consider for instance the following Fig. 4 which shows the
graph of $ln\lvert B_{3}\left(\frac{1}{2}+it\right)\rvert$ for $0\leq t\leq
150$:

Figure 4.  Graph of
$ln\lvert\widetilde{B}_{3}\left(\frac{1}{2}+it\right)\rvert$ for $0\leq t\leq
150$.

In Fig. 4 the region of the first _negative_ zeros
$t^{k}_{m}=\widetilde{t}^{-}_{k}(m)$ for $m=1,...,9$ is marked in orange and
the region of the _positive_ zeros $t^{k}_{m}=\widetilde{t}^{+}_{k}(m-2k^{2})$
is marked in blue. In view of the above, we also obtain the following
alternative description of the zeros of $B_{k}(s)$ in the relevant region,
which does not involve the Lambert function:

######  Proposition 3.2 (alternative description of the zeros of $B_{k}(s)$).

The zeros of $B_{k}(s)$ in the region $2N\leq t\leq 2(N+1)$ are given by

$\widetilde{T}_{j}^{k}=\frac{8N+8\pi\cdot j-11\pi}{4\left(ln(N)-ln(\pi\cdot
k^{2})\right)}$

for

$\frac{N}{\pi}ln\left(\frac{N}{\pi\cdot k^{2}e}\right)+\frac{11}{8}\leq
j\leq\frac{N}{\pi}ln\left(\frac{N}{\pi\cdot
k^{2}e}\right)+\frac{1}{\pi}ln\left(\frac{N}{\pi\cdot
k^{2}}\right)+\frac{11}{8}$ (30)

###### Proof.

In view of Eq. 25, in the region $2N\leq t\leq 2(N+1)$ the following
approximation holds

$ln\left|B_{k}\left(\frac{1}{2}+it\right)\right|\approx\left|\frac{2}{\sqrt{k}}sin\left(\frac{t}{2}ln\left(\frac{N}{\pi
k^{2}}\right)+\frac{11\pi}{8}-N\right)\right|.$ (31)

The zeros of the approximating function are given by

$\widetilde{T}_{j}^{k}=\frac{8N+8\pi\cdot j-11\pi}{4\left(ln(N)-ln(\pi\cdot
k^{2})\right)}$ (32)

for $j\in\mathbb{Z}$. ∎

In particular, let us define the distance between two consecutive zeros of
$B_{k}(s)$ in the region $2N\leq t\leq 2(N+1)$ by

$d(N,k):=\left|\frac{2\pi}{ln(N)-ln(\pi\cdot k^{2})}\right|$ (33)

Let us conclude this section with the following two remarks:

######  Remark 3.1 (The Franca-LeClair approach for RH).

In [6, 7] Franca and LeClair further (formally) introduce the equation

$\frac{t}{2\pi}\cdot ln\left(\frac{t}{2\pi
e}\right)+\frac{1}{\pi}lim_{\delta\rightarrow
0^{+}}\left(arg\left(\zeta\left(\frac{1}{2}+\delta+it\right)\right)\right)=n-\frac{11}{8}.$
(34)

Note that the leading term of this equation is

$\frac{t}{2\pi}\cdot ln\left(\frac{t}{2\pi e}\right)=n-\frac{11}{8},$ (35)

which admit the unique solution $t_{n}^{0}$ for any $n$, as defined in Eq. 29.
In contrast, the full Eq. 34 is only formally defined due to the fact that the
term

$\frac{1}{\pi}lim_{\delta\rightarrow
0^{+}}\left(arg\left(\zeta\left(\frac{1}{2}+\delta+it\right)\right)\right)$
(36)

is not nessecerally well defined for all zeros of $\zeta(s)$. Franca and
LeClair show that if Eq. 36 is well defined then equation Eq. 35 has a
solution $t_{n}$ for any $n$, and to any such solution corresponds a zero
$\rho_{n}=\frac{1}{2}+it_{n}$ of $\zeta(s)$ on the critical line. In
particular, it is shown that the RH is equivalent to the question of the well-
defindness of the term Eq. 36.

It should be noted that the question of the well-defindness of the term Eq. 36
is a highly non-trivial and elusive matter by itself and, as for RH, to which
it is equivalent, any substantial reason for the well-defindness of the term
Eq. 36 is currently lacking. Moreover, in [6, 7] Franca and LeClair also study
the analogous equation for the Davenport-Heilbronn function $\mathcal{D}(s)$,
which is an $L$-function satisfying a functional equation but for which the RH
fails, and as a result for which the analogous version of Eq. 36 is not well
defined for all zeros. The results of [6, 7], however, do not offer insight on
the reason for the difference between $\zeta(s)$ and $\mathcal{D}(s)$.

On the other hand, the results of Franca-LeClair do imply the important fact
that the RH is equivalent to showing that to any zero of $1+\chi(s)$
corresponds a unique zero of $\zeta(s)$ on the critical line, whose imaginary
part is given as a solution of Eq. 34. In the setting of Franca-LeClair, the
relation between the zeros of the functions is a formal matter of adding the
term in Eq. 36 (which is a-priori not well defined) to Eq. 35. However, again,
the results of [6, 7] do not give insight as of why such a relation between
the zeros of $\chi(s)$ and $\zeta(s)$ should exist in practice.

######  Remark 3.2 (Gram points).

Recall that the Riemann-Siegel theta function is defined by

$\nu(t)=arg\left(\Gamma\left(\frac{1}{4}+\frac{it}{2}\right)\right)-\frac{t}{2}ln(t).$
(37)

The $n$-th Gram point is given as the unique solution of the equation
$\nu(g_{n})=\pi n$, see for instance [4]. The Gram points could be
approximated by

$g_{n}\approx\frac{(8n+1)\pi}{4W_{0}(\frac{8n+1}{8e})}.$ (38)

From this, and the definition of $t^{0}_{n}$ in Eq. 29 Spira’s observation
that a Gram point $g_{n}$ is always found between any two consecutive elements
of $t^{0}_{n}$ (and vice versa), follows immediately.

The first few Gram points were computed by Gram in [8] where he also observed
that _typically_ one has ”Gram’s law”:

$Re\left(\zeta\left(\frac{1}{2}+it\right)\right)=(-1)^{n}Z(g_{n})>0,$ (39)

where $Z(t):=e^{i\nu(t)}\zeta\left(\frac{1}{2}+it\right)$ is the Riemann-
Siegel function. As $Z(t)$ is a real function, whenever Eq. 39 is satisfied
for two consecutive Gram points, $g_{n}$ and $g_{n+1}$, it implies the
existence of a zero of zeta between these two point, on the critical line.
However, in [9] Hutchinson computed the Gram points and the values $Z(g_{n})$
up to $n=138$. In particular, Hutchinson found examples in which violations of
Eq. 39 occurs, the first such example occurring for $n=126$. In the next
Section 4 we will present a new interpretation of Gram’s law in terms of the
zeros of $\zeta_{N}(s)$.

##  4\. On the change of zeros of $\zeta_{N}(s)$ with respect to $N$ and their
collisions

In the previous section we described the zeros of
$\zeta_{1}(s)=2\widetilde{\zeta}_{0}(s)$ on the critical strip, and saw that
they all lie on the critical line. In this section we are interested in
studying how the zeros of $\zeta_{N}(s)$ (or $\widetilde{\zeta}_{N}(s)$444All
arguments of this section apply equally well for the accelerated sections
$\widetilde{\zeta}_{N}(s)$.) change with respect to $N$.

First recall that by Rouche’s theorem, for any two complex-valued functions
$F(s)$ and $G(s)$ holomorphic inside some region $K\subset\mathbb{C}$ with
closed contour $\partial K$, if $\lvert G(s)\rvert<\lvert F(s)\rvert$ on
$\partial K$, then $F(s)$ and $F(s)+G(s)$ have the same number of zeros, with
multiplicity, inside $K$. In our case, Let us set

$\begin{array}[]{ccc}F_{N}(s)=\zeta_{1}(s)&;&G_{N}(s)=\sum_{n=2}^{N}B_{n}(s),\end{array}$
(40)

such that $\zeta_{N}(s)=F_{N}(s)+G_{N}(s)$, by definition. It is easy to show
that in compact regions with $\lvert\sigma\rvert>>0$ big enough, the condition
$\lvert G_{N}(s)\rvert<\lvert F_{N}(s)\rvert$ is satisfied. Hence, we have:

  * •

The zeros of $\zeta_{1}(s)$ in the critical strip are in one-to-one
correspondence with the zeros of the sections $\zeta_{N}(s)$, for any $N$, and
no new zeros in the critical strip can be created in the transition from
$\zeta_{1}(s)$ to $\zeta_{N}(s)$.

Moreover, it is expected that the addition of the error term of the AFE
(especially in the transformed case) would also not create new zeros in the
transition from $\widetilde{\zeta}_{N}(s)$ to $\zeta(s)$ in the region $2N\leq
t\leq 2(N+1)$. Hence we get:

  * •

The non-trivial zeros of the Riemann zeta function $\zeta(s)$ on the critical
strip are in one-to-one correspondence with the zeros of the section
$\zeta_{N}(s)$ for $N=\left[\frac{t}{2}\right]$ and the distance between them
is exponentially small with respect to the size of $t$.

It should be noted that this argument does not yet imply anything regarding
the location of the zeros of $\zeta(s)$, but only on the existence of a one-
to-one correspondence between the non-trivial zeros of $\zeta(s)$ and those of
the sections $\zeta_{N}(s)$ from zero to $\left[\frac{t}{2}\right]$. In
summary, in view of the above remark, we obtain:

  * •

The RH would follow from showing that to any zero of
$\widetilde{\zeta}_{0}(s)$ on the critical line corresponds a non-trivial zero
of $\zeta_{N}(s))$ for $N=\left[\frac{t}{2}\right]$ lying _on the critical
line_.

In other words

  * •

Spira’s RH for sections $\Rightarrow$ RH.

Before proceeding let us make the following remark:

######  Remark 4.1 (Comparison to the approach of Franca-LeClair).

As mentioned in the previous section, in the setting of Franca-LeClair the
zeros of $1+\chi(s)$ arise as leading term approximation of the full Franca-
LeClair equation (Eq. 34), which should theoretically coincide (assuming RH)
with the zeros of $\zeta(s)$ on the critical line. In particular, this
suggested theoretical relation is the result of the algebraic fact that Eq. 35
is the first-order approximation of Eq. 34. In [6, 7] Franca and LeClair also
presented extensive numerical evidence for various connections between the
properties of the zeros of $1+\chi(s)$ and those of the known zeros of
$\zeta(s)$. However, Franca and LeClair do not suggest further formal
explanations for why a connection between the two collections exists.

In our setting of the AFE the collection of zeros of $1+\chi(s)$ arises as the
zeros of the zero-th section $\widetilde{\zeta}_{0}(s)$. However, in our
setting we also get the corresponding collection zeros of the section
$\widetilde{\zeta}_{N}(s)$, for any other $N\in\mathbb{N}$. In this sense the
zeros of $\widetilde{\zeta}_{N}(s)$ could be thought of as a generalization of
the Franca-LeClair approximations for any $N$. In particular, we view these
collections as forming a gradual ”bridge” between the zeros of $1+\chi(s)$ and
the actual zeros of $\zeta(s)$ essentially attained for
$N=\left[\frac{t}{2}\right]$, which gives a satisfying explanation for why the
relation between the FL zeros and those of $\zeta(s)$ exists.

We are hence interested in studying the way the zeros change with respect to
$N$. As the parameter $N$ is discrete, we need to make sense of what we mean
by ”change with respect to $N$”. For any $N\in\mathbb{N}$ define

$\zeta^{t}_{N}(s)=(1-t)\cdot\zeta_{N}(s)+t\cdot\zeta_{N+1}(s),$ (41)

for $0\leq t\leq 1$. In particular, $\zeta^{0}_{N}(s)=\zeta_{N}(s)$ and
$\zeta^{1}_{N}(s)=\zeta_{N+1}(s)$. Let us consider the way the zeros of
$\zeta_{N}^{t}(s)$ change with respect to $t$. Assume the zeros of
$\zeta_{N}(s)$ in the critical strip are given by $\rho^{N}_{n}$ for
$n\in\mathbb{Z}$. For any $n\in\mathbb{Z}$ we can define the continuous family
of zeros $\rho_{n}^{N}(t)$ of $\zeta^{t}_{N}(s)$. This can be done only as
long as no double zeros occur. If a double zero occurs we refer to such an
instance as a ”collision” between zeros. Starting from $\zeta_{1}(s)$ we can
thus inductively define $\rho_{n}^{N}$ for any $\zeta_{N}(s)$ (similarly
$\widetilde{\rho}_{n}^{N}$ for any $\widetilde{\zeta}_{N}(s)$) as follows:

  1. (1)

As long as no collisions occur define inductively
$\rho_{n}^{N+1}=\rho_{n}^{N}(1)$. The main feature is that collisions can
occur only between two consecutive zeros $\rho_{n}^{N}$ and $\rho_{n+1}^{N}$
and as long as no collisions occur the zeros must remain on the critical line.

  2. (2)

If a collision between two consecutive zeros $\rho_{n}^{N}(t)$ and
$\rho^{N}_{n+1}(t)$ on the critical line occurs for some $0\leq t_{0}\leq 1$
then the two zeros get ”pushed off the critical line” in a symmetric manner
along the critical line, so that we can continue to define $\rho_{n}^{N}(t)$
and $\rho^{N}_{n+1}(t)$ for $t_{0}\leq t\leq 1$ such that
$\rho_{n}^{N}(t)=1-\rho^{N}_{n+1}(t)$.

  3. (3)

After two consecutive zeros collided, for some $N$ and $t$, the only way a
collision can occur again is if the two zeros return to the critical line.
This is because the only two zeros symmetric along the critical line can meet
along the critical line.

Let us consider the following example:

######  Example 4.2 (Collisions for $\widetilde{\zeta}_{N}(s)$).

The following Fig. 5 illustrates the phenomena of collisions by showing the
graphs of $ln\lvert\zeta(\frac{1}{2}+it)\rvert$ (blue) and
$ln\lvert\widetilde{\zeta}_{N}(\frac{1}{2}+it)\rvert$ for $N=8,9$ (orange) in
the range $86\leq t\leq 90$:

Figure 5. Graphs of $ln\lvert\zeta(\frac{1}{2}+it)\rvert$ (blue) and
$ln\lvert\widetilde{\zeta}_{N}(\frac{1}{2}+it)\rvert$ for $N=8$ (a) and $N=9$
(b) (orange) in the range $86\leq t\leq 90$.

Figure 5 shows that $\widetilde{\zeta}_{8}(s)$ admits two zeros on the
critical line in this range. However, $\widetilde{\zeta}_{9}(s)$ no longer has
zeros on the critical line in this range. The reason is that a collision
between the two zeros occurred in $\widetilde{\zeta}^{t}_{8}(s)$ for a certain
$0\leq t\leq 1$. It should be noted that $\widetilde{\zeta}_{9}(s)$ has two
zeros in the range $86\leq t\leq 90$ given approximately by $0.74+88.12i$ and
$0.25+88.12i$. The zeros of $\widetilde{\zeta}_{N}(s)$ remain off the critical
until $N=22$ where a second collision occurs and the zeros return to the
critical line. This is shown in Fig. 6

Figure 6. Graphs of $ln\lvert\zeta(\frac{1}{2}+it)\rvert$ (blue) and
$ln\lvert\widetilde{\zeta}_{N}(\frac{1}{2}+it)\rvert$ for $N=22$ (a) and
$N=23$ (b) (orange) in the range $86\leq t\leq 90$.

As one can see, even though a collision occurred at $N=8$ and ”pushed the
zeros off” the critical line, an additional complementing collision occurred
at $N=22$ brining the zeros ”back to the critical line”.

In general, the RH would follow if this phenomena holds for any consecutive
pair of zeros. That is:

  * •

The RH would follow if for any pair of zeros the following holds: for any
collision ”pushing the zeros off” the critical line occurring for certain $N$
there must occur a corresponding collision ”pushing the zeros back” to the
critical line for later $N^{\prime}$.

In fact, we will not consider this question directly. Instead, we will suggest
in the next section a method to avoid collisions all together, by changing the
order of summation in the elements of $\zeta_{N}(s)$. Let us conclude this
section with the following two remarks:

######  Remark 4.3 (Non resolved collisions for the classical AFE range of
approximation).

It should be noted that the sections $\zeta_{N}(s)$ approximate zeta in two
different ways: (a) due to the functional equation it approximates $\zeta(s)$
in the region $\sqrt{2\pi N}\leq t\leq 2\pi N$ (which is our main concern).
(b) due to the classical approximate functional equation it approximates
$\frac{1}{2}\zeta(s)$ in the region $2\pi N^{2}\leq t\leq 2\pi(N+1)^{2}$.

It was already observed by Spira that the sections $\zeta_{N}(s)$ might admit
zeros off the critical line in the region $2\pi N^{2}\leq t\leq 2\pi(N+1)^{2}$
(in which according to the AFE it approximates $\frac{1}{2}\zeta(s)$). For
instance, for $N=5$ the section $\zeta_{5}(s)$ admits two zeros off the
critical line in the region $219\leq t\leq 222$. The reason for this is that a
collision occurs at the previous $N=4$ and is not resolved. It should be noted
that the number $N=5$ of sections required for the AFE approximation is
extremely small compared to the number of sections
$N=\left[\frac{t}{2}\right]=110$ required for the approximation of $\zeta(s)$
in this region, which is the approximation we are interested in.

######  Remark 4.4 (A new interpretation of Gram’s law).

In view of Remark 3.2 the elements $t^{0}_{n}$ are the zeros of $\lvert
1+\chi(s)\rvert$, while the Gram points are actually exactly the local maximum
points of the same function on the critical line. Hence, the Gram point could
be considered as the ”middle point” between two zeros of $1+\chi(s)$. Recall
that the $126$-th zero is a zero which violates Gram’s law. Figure 7 shows the
way the elements $t^{N}_{126}=Im(\widetilde{\rho}_{126}^{N})$ (orange) of
$\widetilde{\zeta}_{N}(s)$ change for $0\leq N\leq 89$:

Figure 7. The elements $t^{N}_{126}=Im(\widetilde{\rho}_{126}^{N})$ (orange)
and the Gram point $g_{126}$ (blue) for $0\leq N\leq 89$

As one can see in Fig. 7 during the process of development of the zeros from
zero to $N=89$ the imaginary part eventually crosses the Gram point, that is
the middle point. Hence, in general, ”Gram’s law” can be re-phrased as:

  * •

Gram’s law is the observation that (for each individual $n$) the elements
$t^{N}_{n}=Im(\widetilde{\rho}^{N}_{n})$ do not tend to eventually cross the
Gram point $g_{n}$, that is, the overall distance travelled by $t^{N}_{n}$ is
usually less than half the distance between the intial position of the zeros
for $N=0$.

The following features should be mentioned:

  1. (1)

In Gram’s law the point $g_{n}$ (blue), which is a local maxima of
$\lvert\zeta_{0}(\frac{1}{2}+it)\rvert$, is kept ”static” and does not change
with respect to $N$. If, however, instead of $g_{n}$ one considers the local
maxima of $\lvert\widetilde{\zeta}_{N}(\frac{1}{2}+it)\rvert$ one actually
obtains a statement which is essentially our approach to the RH. It should be
noticed that the persistence of such local maxima between two consecutive
zeros is equivalent to the statement that no collisions occur and their
reappearance is the result of the occurrence of a second collision. In this
sense, our approach to the RH could be considered as a ”dynamic”, more
refined, version of the cruder ”static” Gram’s law.

  2. (2)

As phrased above Gram’s law is a statement about the typical process of
development of an _individual_ zero, that is of the sequence $t^{N}_{n}$ for a
_given individual $n$_. Indeed, that the sequence $t_{n}^{N}$ for a given
individual zero does not tend to pass an overall distance of half the distance
between $t^{0}_{n}$ and $t^{0}_{n+1}$ (recall formula Eq. 33). In our
approach, however, we do not consider the behaviour of an individual zero but
rather study the _mutual development of a pair of consecutive zeros_. In
particular, we are interested in collisions, or lack of collisions, for
_pairs_ of zeros. In this sense, our approach to the RH could be rephrased as
saying that the sequences $t^{N}_{n}$ and $t^{N}_{n+1}$ for a pair of
consecutive zeros _can never_ mutually pass together an overall distance which
exceeds the original distance between $t^{0}_{n}$ and $t^{0}_{n+1}$. In this
sense, RH could considered as a ”twice stronger” statement than Gram’s law.

Let us consider the following example:

######  Example 4.5 (A pair of non-colliding zeros).

Figure 8 shows the consecutive sequences $t^{N}_{132}$ (brown) and
$t^{N}_{133}$ (blue) together with the imaginary parts zeros $t_{132}$ and
$t_{133}$ (left) and the consecutive sequences $\widetilde{t}^{N}_{132}$
(brown) and $\widetilde{t}^{N}_{133}$ (blue) together with the imaginary parts
of the zeros $\rho_{132}$ and $\rho_{133}$ (right). As one can see from Fig.
8, for the presented $132$-th and $133$-th zeros, the sequences $t_{132}^{N}$
and $t_{133}^{N}$ (respectively, $\widetilde{t}_{132}^{N}$ and
$\widetilde{t}_{133}^{N}$) develop separately, and no collisions between the
two occurs for any $N$. As a result, the sequences $\rho_{132}^{N}$ and
$\rho_{133}^{N}$ (respectively, $\widetilde{\rho}_{132}^{N}$ and
$\widetilde{\rho}_{133}^{N}$) remain on the critical line, for any $N$:

Figure 8.  (a) The consecutive sequences $t^{N}_{132}$ (brown) and
$t^{N}_{133}$ (blue) together with $t_{132}$ and $t_{133}$. (b) The
consecutive sequences $\widetilde{t}^{N}_{132}$ (brown) and
$\widetilde{t}^{N}_{133}$ (blue) together with $t_{132}$ and $t_{133}$ for
$N=1,...,150$.

It is obvious that for any such pair of consecutive zeros $\rho_{n}$ and
$\rho_{n+1}$, for which a collision does not occur in their corresponding
sequences, both zeros must remain on the critical line. That is, any such pair
of non-colliding zeros nessecerally satisfy RH. In the next section we would
consider the more ”interesting” case of colliding zeros.

It should also be noted that both the classical $\rho_{n}^{N}$ and transformed
$\widetilde{\rho}_{n}^{N}$ sequences are non-colliding in the case of
$n=132,133$ presented in Fig. 8. This is typical of the general case in which
the classical and transformed sequences are seen to collide$\setminus$non-
collide together. That is the collision property is an essential feature of a
given pair of zeros and not of the specific sequence (classical or
transformed) considered.

##  5\. Avoiding collisions via the repelling re-arrangement of summation

In the previous section we saw an example of non-colliding pair of zeros.
Before explaining how collisions might be avoided let us first consider an
example of colliding zeros.

######  Example 5.1 (A pair of colliding zeros).

The following Fig. 9 shows the consecutive sequences $t^{N}_{725}$ (brown) and
$t^{N}_{726}$ (blue) together with the imaginary parts of the zeros $t_{725}$
and $t_{726}$ (left) and the consecutive sequences $\widetilde{t}^{N}_{725}$
(brown) and $\widetilde{t}^{N}_{726}$ (blue) together with the imaginary parts
of the zeros $\rho_{725}$ and $\rho_{726}$ (right) for $N=1,..,300$:

Figure 9.  (a) The consecutive sequences $t^{N}_{725}$ (brown) and
$t^{N}_{726}$ (blue) with $t_{725}$ (green) and $t_{726}$ (red). (b) The
consecutive sequences $\widetilde{t}^{N}_{725}$ (brown) and
$\widetilde{t}^{N}_{726}$ (blue) with $t_{725}$ (green) and $t_{726}$ (red)
for $N=1,...,300$.

As one can see from Fig. 9, for the presented $725$-th and $726$-th zeros, the
sequences $t_{725}^{N}$ and $t_{726}^{N}$ (respectively,
$\widetilde{t}_{725}^{N}$ and $\widetilde{t}_{726}^{N}$) do collide for
certain values of $N$. However, as expected, after this collision the two
zeros eventually part and continue on separate paths.

Note that the development of the sequences $t_{n}^{N}$ and
$\widetilde{t}^{N}_{n}$ bare some clear resemblance to the partial sums
$S_{N}(s)$ and $\widetilde{S}_{N}(s)$, as discussed in Section 2, compare for
instance Fig. 2. Indeed, we see that the sequences $t_{n}^{N}$ and
$\widetilde{t}^{N}_{n}$ fluctuate around various values before stabilizing at
the limit $t_{n}$ for $N>>0$. In fact, like in the case of the partial sums
$S_{N}(s)$ the transitions in the sequence $t_{n}^{N}$ occur in the regions
$\left[\frac{t}{2(M+1)\pi}\right]\leq t\leq\left[\frac{t}{2M\pi}\right]$. For
instance the two collisions for $t_{n}^{N}$ in Fig. 9 occur at the following
intervals:

  1. (1)

The first collision occurs around the interval
$\left[\frac{t}{24\pi}\right]\leq N\leq\left[\frac{t}{12\pi}\right]$

  2. (2)

The second collision occurs around the interval
$\left[\frac{t}{4\pi}\right]\leq N\leq\left[\frac{t}{2\pi}\right]$.

We expect that up until around the half of these intervals the two zeros are
”attracted towards each other” and afterwards, in the second half of the
interval, get ”repelled away from each other”. According to this viewpoint,
let us consider the following repelling re-arrangements of the indices

$\begin{array}[]{ccc}R(n):=\left\\{\begin{array}[]{cc}n&n\leq 13\\\
41-n&13<n<28\\\ n&28\leq n\leq 85\\\ 256-n&85<n<171\\\ n&171\leq
n\end{array}\right.&;&\widetilde{R}(n):=\left\\{\begin{array}[]{cc}n&n\leq
30\\\ 84-n&30<n<54\\\ n&54\leq n\leq 177\\\ 504-n&177<n<327\\\ n&327\leq
n\end{array}\right..\end{array}$ (42)

The idea behind defining $R(n)$ and $\widetilde{R}(n)$ in such a manner is
that we want to add the ”repelling” elements before ”attracting” ones out of
an expectation that this would lead to a cancellation of the collisions. In
particular, let us define the re-arranged sections

$\begin{array}[]{ccc}\zeta_{N}(s;R):=\sum_{n=1}^{N}B_{R(n)}(s)&;&\widetilde{\zeta}_{N}(s;\widetilde{R}):=\sum_{n=1}^{N}\widetilde{B}_{\widetilde{R}(n)}(s).\end{array}$
(43)

Set $\rho_{n}^{N}(R)$ and $\widetilde{\rho}_{n}^{N}(\widetilde{R})$ for the
zeros of $\zeta_{N}(s;R)$ and $\widetilde{\zeta}_{N}(s;\widetilde{R})$ and
denote by $t_{n}^{N}(R)$ and $\widetilde{t}_{n}^{N}(\widetilde{R})$ the
corresponding imaginary parts. Consider Fig. 10:

Figure 10.  (a) The rearranged sequences $t^{N}_{725}(R)$ (brown) and
$t^{N}_{726}(R)$ (blue) with $t_{725}$ (green) and $t_{726}$ (red). (b) The
rearranged sequences $\widetilde{t}^{N}_{725}(\widetilde{R})$ (brown) and
$\widetilde{t}^{N}_{726}(\widetilde{R})$ (blue) with $t_{725}$ (green) and
$t_{726}$ (red) for $N=1,...,300$.

Figure 10 shows us is that, contrary to Fig. 9 for the ordinary sections, for
the rearranged sections $\zeta_{N}(s;R)$ and
$\widetilde{\zeta}_{N}(s;\widetilde{R})$ the collisions have been altogether
avoided. In other words, the collisions seen in Fig. 9 are not an essential
feature of the zeros but rather a by-product of the trivial order of summation
considered. Of course, whenever a collision can be cancelled it implies that
the pair of zeros satisfy RH.

We conjecture that the procedure described above holds in general and can be
applied to any consecutive pair of zeros. Concretely, we conjecture:

  * •

For any pair of consecutive zeros, collisions in the sequences $t_{n}^{N}$ and
$t_{n+1}^{N}$ (or $\widetilde{t}_{n}^{N}$ and $\widetilde{t}_{n+1}^{N}$) can
occur only along intervals of the form $\left[\frac{t}{2M_{1}\pi}\right]\leq
N\leq\left[\frac{t}{2M_{2}\pi}\right]$. A collision can always be cancelled by
rearranging the order of summation of the elements $B_{n}(s)$
($\widetilde{B}_{n}(s)$) to be in reverse order along such an interval of
collision.

Let us conclude this section with the following remarks:

######  Remark 5.2 (The chaotic region).

Let us note that intervals of the form

$\left[\frac{t}{2(M+1)\pi}\right]\leq N\leq\left[\frac{t}{2M\pi}\right]$ (44)

become smaller as $M$ grows and after some $M_{0}=M_{0}(t)$ become essentially
trivial. Thus the range of existence of the regulated intervals only begins
after an initial region of the form $N\leq\left[\frac{t}{2M_{0}\pi}\right]$,
to which we refer as the ”chaotic region” of $t$. In particular, our
conjecture includes the assumption that collisions cannot occur at all during
the initial chaotic region but rather only when the intervals Eq. 44 start
becoming regulated, that is, big enough. We can rephrase by saying that the
initial chaotic region does not have enough ”energy” to bring the zeros into
collision or to travel a mutual distance of more than $\lvert
t^{0}_{n+1}-t^{0}_{n}\rvert$, compare Eq. 33.

######  Remark 5.3 (Non-cancellation of collision for the Davenport-Heilbronn
function).

Recall that the Davenport-Heilbronn functions are a class of Dirichlet
functions which satisfy a functional equation but for which RH fails, that is
for which there exist zeros off the critical line, see [2, 25]. Consider the
function

$\mathcal{D}(s)=\frac{(1-i\kappa)}{2}L(s,\chi_{5,2})+\frac{(1+i\kappa)}{2}L(s,\overline{\chi}_{5,2})$
(45)

with

$\kappa=\frac{\sqrt{10-2\sqrt{5}}-2}{\sqrt{5}-1}.$ (46)

The functional equation for the Davenport-Heilbronn function $\mathcal{D}(s)$
is given by

$\xi(s)=\xi(1-s),$ (47)

where

$\xi(s)=\left(\frac{\pi}{5}\right)^{-\frac{s}{2}}\Gamma\left(\frac{1+s}{2}\right)\mathcal{D}(s).$
(48)

It should be noted that contrary to $\zeta(s)$ the function $\mathcal{D}(s)$
does not have an Euler product. If we apply Euler acceleration of series to
$\mathcal{D}(s)$ we can express

$\mathcal{D}(s)=\sum_{n=0}^{\infty}\widetilde{A}_{DH}(n,s),$ (49)

where

$\widetilde{A}_{DH}(n,s):=\frac{1}{2^{n+2}}\sum_{k=0}^{n}(-1)^{k+1}\binom{n}{k}\frac{(1-i\kappa)\chi_{5,2}(k)+(1+i\kappa)\overline{\chi}_{5,2}(k)}{k^{s}}.$
(50)

In view of the functional equation, we define

$\widetilde{\xi}_{N}(s)=\sum_{n=0}^{N}\widetilde{B}_{DH}(n,s),$ (51)

where

$\widetilde{B}_{DH}(n,s):=\frac{1}{2}\left[\left(\frac{\pi}{5}\right)^{-\frac{s}{2}}\Gamma\left(\frac{1+s}{2}\right)\widetilde{A}_{DH}(n,s)+\left(\frac{\pi}{5}\right)^{\frac{s-1}{2}}\Gamma\left(\frac{2-s}{2}\right)\widetilde{A}_{DH}(n,1-s)\right]$
(52)

We mention (without proof) that $\widetilde{X}_{N}(s)$ attains exponential
accuracy of $\xi(s)$ around $N=N(s)=\left[2t\right]$ in this case.

Franca and LeClair show in [6, 7] that the zeros
$\rho^{DH}_{0,n}=\frac{1}{2}+it^{DH}_{0,n}$ of $\widetilde{\xi}_{0}$ all lie
on the critical line their imaginary part $t^{DH}_{0,n}$ can be expressed in
terms of the Lambert function as follows

$t^{DH}_{0,n}:=\frac{2\pi(n-\frac{5}{8})}{W_{0}(5e^{-1}(n-\frac{5}{8}))}.$
(53)

The following Fig. 11 shows (a) the consecutive sequences $t^{DH}_{N,44}$
(brown) and $t^{DH}_{N,45}$ (blue) with $t^{DH}_{44}$ (green) and
$t^{DH}_{45}$ (red). (b) The consecutive sequences $\widetilde{t}^{DH}_{N,44}$
(brown) and $\widetilde{t}^{DH}_{N,45}$ (blue) with $t^{DH}_{44}$ (green) and
$t^{DH}_{45}$ (red) for $N=1,...,200$.

Figure 11.  (a) The consecutive sequences $t^{DH}_{N,44}$ (brown) and
$t^{DH}_{N,45}$ (blue) with $t^{DH}_{44}$ (green) and $t^{DH}_{45}$ (red). (b)
The consecutive sequences $\widetilde{t}^{DH}_{N,44}$ (brown) and
$\widetilde{t}^{DH}_{N,45}$ (blue) with $t^{DH}_{44}$ (green) and
$t^{DH}_{45}$ (red) for $N=1,...,200$.

In contrast to the zeta function, the the collision of the sequences presented
in Fig. 11 is essential and cannot be cancelled by changing the order of
summation. Moreover, the collision occurs at $N=12$, that is, already within
the initial _chaotic region_. In other words, it seems that in the Davenport-
Heilbronn setting the chaotic region does have enough _energy
$\setminus$momentum_ to collide consecutive zeros. In particular, the zeros of
the DH function $\mathcal{D}(s)$ off the critical line seem to present an
essentially different behaviour then those of the zeta function $\zeta(s)$.

######  Remark 5.4 (Montgomery’s pair correlation conjecture).

For pairs of zeros of zeta one has (assuming RH) the famous pair correlation
conjecture due to Montgomery, see [18]. Let $\alpha\leq\beta$ and set

$A(T;\alpha,\beta):=\left\\{(\rho,\rho^{\prime})\mid
0<\rho,\rho^{\prime}<T\textrm{ and
}\frac{2\pi\alpha}{ln(T)}\leq\rho-\rho^{\prime}\leq\frac{2\pi\beta}{ln(T)}\right\\}.\
$ (54)

The conjecture states that

$N(T;\alpha,\beta):=\sum_{A}1\sim\left(\int_{\alpha}^{\beta}\left(1-\frac{sin(\pi
u)}{\pi u}\right)du+\delta_{0}([\alpha,\beta])\right)\frac{T}{2\pi}ln(T)$ (55)

for $T\rightarrow\infty$. Since the integral is small when $u$ is small, the
conjecture is typically intuitively understood as expressing the idea that
consecutive zeros ”repel each other”. Hence, although the term repulsion
suggests a dynamic relation, in the context of the Montgomery conjecture the
notion of ”repulsion” is a statistical one.

In our setting we also speak about ”repulsion” between consecutive zeros of
zeta or, more concretely, between the consecutive sequences $t_{n}^{N}$ and
$t_{n+1}^{N}$. However, in our setting the notion of repulsion is indeed a
dynamic one rather than statistic. Concretely, we say that at stage $N$ the
sequence is ”repelling” if the distance between the two sequences decreases

$\lvert t_{n+1}^{N}-t_{n}^{N}\rvert<\lvert t_{n+1}^{N+1}-t_{n}^{N+1}\rvert$
(56)

and ”attracting” if the opposite inequality occurs and the zeros get closer to
one another. Note that the repulsion or attraction of adding $B_{n}(s)$
depends on the position of the zeros of $B_{n}(s)$ (described in Proposition
3.1) relative to the zeros of the section to which they are added. Our
approach is that for any pair of consecutive zeros the overall ”attraction”
cannot exceed the original distance $\lvert t^{0}_{n+1}-t^{0}_{n}\rvert$.

##  6\. Summary and Concluding Remarks

The Riemann hypothesis is the postulate that all non-trivial zeros of
$\zeta(s)$ lie on the critical line. Riemann himself computed the first three
zeros of $\zeta(s)$ (as communicated by Siegel [22]). As of today, the Riemann
hypothesis has been numerically verified for zeros $\rho$ with $Im(\rho)$ up
to around $3\cdot 10^{12}$, see [19]. However, since its introduction a
plausibility argument for RH, that is a conceptual reason for its validity,
aside from numerical verification, has been essentially thought-after555For
instance Edwards writes as follows in his classical book [4]: _”Even today,
more than a hundred years later, one cannot really give any solid reasons for
saying that the truth of the RH is ”probable” etc. Also the verification of
the hypothesis for the first three and a half million roots above the real
axis perhaps makes it more ”probable”. However, any real _reason_ , any
plausibility argument or heuristic basis for the statement, seems entirely
lacking” (H. M. Edwards, 1974)_.

In this work we suggested a new approach for the study of the zeros of the
zeta function $\zeta(s)$ by studying the dynamic way in which the zeros of the
section $\zeta_{N}(s)$ change with respect to $N$. The zeros of
$\zeta_{1}(s)$, which are regulated and well-understood, start on the critical
line and then move their position as $N$ changes. A pair of zeros can go off
the critical line only if a collision between the two zeros occur. We
conjectured, based on numerical evidence, that collisions could always be
avoided by changing the order of summation of the elements $B_{n}(s)$
comprising the section $\zeta_{\left[\frac{t}{2}\right]}(s)$, in a specific
manner to which we refer as a repelling re-arrangement.

In this sense we suggest that the observed dynamic repulsion phenomena could
be viewed as a certain ”plausibility argument” for RH. Indeed, the idea
suggested in this work, that the zeros begin on the critical line and then
develop in a way that dynamically repels them for colliding, obliges them to
stay on the critical line, which at the end of the process is essentially the
RH. Of course, as explained, if it would be possible to prove that the
introduced repelling re-arrangements indeed avoid collisions, in general, for
any pair of consecutive zeros, this would imply RH.

## References

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# Hybrid Quantum Neural Network Advantage for Radar-Based Drone Detection and
Classification in Low Signal-to-Noise Ratio

Aiswariya Sweety Malarvanan

###### Abstract

In this paper, we investigate the performance of a Hybrid Quantum Neural
Network (HQNN) and a comparable classical Convolution Neural Network (CNN) for
detection and classification problem using a radar. Specifically, we take a
fairly complex radar time-series model derived from electromagnetic theory,
namely the Martin-Mulgrew model, that is used to simulate radar returns of
objects with rotating blades, such as drones. We find that when that signal-
to-noise ratio (SNR) is high, CNN outperforms the HQNN for detection and
classification. However, in the low SNR regime (which is of greatest interest
in practice) the performance of HQNN is found to be superior to that of the
CNN of a similar architecture.

###### keywords:

Drone Classification , Radar Detection , Convolutional Neural Networks ,
Hybrid Quantum Neural Networks

††journal:

[first]organization=Department of Physics, University of
Cologne,addressline=Albertus-Magnus-Platz, city=Cologne, postcode=50923,
state=NRW, country=Germany

##  1 Introduction

Machine learning has been successfully applied to a wide variety of problems.
Quantum Machine Learning (QML) is an emerging field that merges the principles
of quantum computing with classical machine learning techniques. The aim is to
take advantage of quantum phenomena to enhance the efficiency and capabilities
of classical machine learning algorithms [1] [2] [3].

Hybrid quantum neural networks (HQNNs), which leverages classical deep
learning architectures alongside quantum algorithms, such as Variational
Quantum Circuits (VQCs), to process data, stands out as a significant research
domain in QML [4] [5] [6].

The use of HQNNs has been explored for a diverse set of problems including
drug response prediction [7], detection of dementia [8], multi-classification
tasks [9] [10], remote sensing imagery classification [11], detection of
cyberattacks [12], gender recognition from facial images [13], and for the
classification of finance and MNIST data [14] [15].

The widespread use of drones across various sectors has motivated the need for
robust methods for their reliable detection and classification. A major
motivation for this is that the statistical model of radar returns is complex,
and in many cases analytical solutions are not available. The use of machine
learning and deep learning for this problem has been investigated in some
prior work [16] [17] [18] [19].

In this article, we use an HQNN with parallel quantum dense layers for the
drone detection and classification problem. Previous work had investigated the
detection and identification of drones using CNN [20] [19]. This work extends
the analysis to HQNN and compares the performance of the HQNN with a CNN of a
comparable architecture. Specifically, as in the prior work, we employ radar
micro-Doppler signatures obtained from the Short-Time Fourier Transform (STFT)
spectrograms of the time-series data of the radar signals reflected from the
drones, simulated based on the Martin-Mulgrew (MM) model. We study the data in
X-band radar simulation scenarios. We examine the binary classification of
drones versus noise for radar detection. We also compare the performance of
HQNN models with classical CNNs for the multi-class classification problem. We
find an unambiguous gain from using HQNNs in the low-SNR regime, the regime of
greatest practical interest. The HQNN is found to perform better for lower SNR
compared to a CNN of a comparable architecture. The results strongly suggest
that HQNNs should be explored for the application to radar time series
analysis.

In Section 2, we review the radar model used in the paper, including the time
domain model, the frequency domain representation, and the proposed detection
and classification methodology.

In Section 3, the data generation method is presented, and the neural
architecture is discussed, including CNN and HQNN. The model training approach
in presented in Section 3.3. The results are presented in Section 4, followed
by the conclusion in Section 5.

##  2 Background

###  2.1 Radar Signal Model: Time Domain Model

The Martin-Mulgrew (MM) model [21], originally designed to analyze radar
returns of aircraft propeller blades, can model radar return signals (in the
form of complex time series of desired time duration) of any aerial object
with rotating propellers, such as drones. This MM model is given as:

$\displaystyle\Psi(t)=$ $\displaystyle A_{r}e^{j\left(2\pi
f_{c}t-\frac{4\pi}{\lambda}\left(R+V_{rad}t\right)\right)}$
$\displaystyle\sum_{n=0}^{N-1}\left(\alpha+\beta\cos\left(\Omega_{n}\right)\right)e^{-j\frac{L_{1}+L_{2}}{2}\gamma_{n}}\text{sinc}\left(\frac{L_{2}-L_{1}}{2}\gamma_{n}\right)$

where

$\displaystyle\alpha$
$\displaystyle=\sin\left(|\theta|+\Phi_{p}\right)+\sin\left(|\theta|-\Phi_{p}\right)$
(1) $\displaystyle\beta$
$\displaystyle=\text{sign}\left(\theta\right)\left(\sin\left(|\theta|+\Phi_{p}\right)-\sin\left(|\theta|-\Phi_{p}\right)\right)$
$\displaystyle\Omega_{n}$
$\displaystyle=2\pi\left(f_{rot}t+\frac{n}{N}\right)$
$\displaystyle\gamma_{n}$
$\displaystyle=\frac{4\pi}{\lambda}\cos\left(\theta\right)\sin\left(\Omega_{n}\right)$

  *   * $A_{r}$ is a real, scaling factor,

  * $L_{1}$ is the distance of the blade roots from the centre of rotation,

  * $L_{2}$ is the distance of the blade tips from the center of rotation,

  * $N$ is the number of blades,

  * $R$ is the range of the target,

  * $V_{rad}$ is the radial velocity of the center of rotation with respect to the radar,

  * $\lambda$ is the wavelength of the transmitted radar signal,

  * $\theta$ is the angle between the plane of rotation and the line of sight from the radar to the center of rotation,

  * $f_{c}$ is the frequency of the transmitted radar signal,

  * $f_{rot}$ is the frequency of rotation of drone blades,

  * $t$ is the time duration of radar returns,

  * $\Phi_{p}$ is the pitch of the blades

The simulation parameters $N$, $L_{1}$, $L_{2}$, and $f_{rot}$ characterises
drone blades/propellers while $\lambda$ and $f_{c}$ determine the type of
radar. The position of the drone relative to the radar is defined by $\theta$,
$\Phi_{p}$, $R$ and $V_{rad}$. The data generated using the MM model is
discretized at a sampling frequency of $f_{s}$.

An important parameter of interest is the single pulse signal-to-noise ratio,
which is defined as $10\log_{10}A_{r}^{2}/\sigma_{n}^{2}$, where
$\sigma_{n}^{2}$ is the noise variance. Note that coherent processing of
$N_{\text{pulses}}$ pulses will give an additional gain of
$10\log_{10}N_{\text{pulses}}$. Also note that since the model focuses on the
radar returns from the rotating blades, the SNR under consideration could be
substantially lower than that of the body (e.g., rotating blades on a drone
relative to the drone body). For simplicity, by SNR we mean single-pulse SNR,
unless specified otherwise.

In this work, radar returns of an X-band radar from five different types of
commercially available drones (DJI Mavic Air 2, DJI Mavic Mini, DJI Matrice
300 RTK, DJI Phantom 4 and Parrot Disco) are simulated using the MM model. The
simulated X-band radar has a wavelength $\lambda$ of 3 cm, a transmitting
frequency $f_{c}$ of 10 GHz, and a pulse repetition frequency (PRF) of 10kHz.
The approximate drone parameters of each drone for which the radar returns are
simulated are shown in Table 1.

Drone | N |  $L_{1}$ [cm] |  $L_{2}$ [cm] |  $f_{r}$ [Hz]
---|---|---|---|---
DJI Mavic Air 2 | 2 | 0.50 | 7.00 | 91.66
DJI Mavic Mini 2 | 2 | 0.50 | 3.50 | 160.00
DJI Matrice 300 RTK | 2 | 5.00 | 26.65 | 70.00
DJI Phantom 4 | 2 | 0.60 | 5.00 | 116.00
Parrot Disco | 2 | 1.00 | 10.40 | 40.00
Table 1: Approximate drone parameters of five drones used in this study

###  2.2 Frequency Domain Representation

The radar model time series is time-varying and nonstationary. As a result,
frequency is constant and well defined only in a short time window. Hence, the
short-time Fourier transform (STFT) is utilized to obtain the frequency
spectrum.

Figure 1 shows a time series plot (real and imaginary separately) for a
particular drone type, namely the Parrot Disco. The corresponding real and
imaginary parts of the STFT of the time domain signal are shown in Figure 2,
in the absence of noise. Blade flashes are clearly visible and appear at
periodic time intervals, as expected. The STFTs for SNR -5 dB is shown in Fig.
3. The blade flashes are less evident, as expected, though there is some
residual structure. Figure 4 shows the STFTs for the same SNR for a different
type of drone, namely the DJI Matrice 300 RTK. There are some differences
between this figure and the previous one, thus making the utility of ML for
detection and classification plausible. Figure 5 shows a sample time series of
additive Gaussian white noise. A corresponding STFT produced using noise for
both real and imaginary inputs is shown in Figure 6. When the SNR is very low,
it is expected that the STFT of any drone will be almost indistinguishable
from that of the additive Gaussian noise.

Figure 1: Real and imaginary valued portions of the time series for drone
Parrot Disco, produced using the MM model. The signal contains no noise.
Figure 2: Real and imaginary valued portions of the STFT of drone Parrot
Disco, produced using the MM model. The signal contains no noise. Figure 3:
Real and imaginary valued portions of the STFT for drone Parrot Disco,
produced using the MM model, with SNR -5. Figure 4: Real and imaginary valued
portion of the STFT for drone DJI Matrice 300 RTK, produced using the MM
model, with SNR -5. Figure 5: Time series of gaussian white noise.  Figure 6:
STFT of gaussian white noise.

As shown in prior work, when the time window is short (relative to the blade
rotation rate) and the radar PRF is high enough, the spectrogram reveals the
blade flashes. At the other end, when the time window is long, it has been
shown analytically (from the Martin-Mulgrew model) as well as experimentally
that the resulting STFT has Helicopter Rotor Modulation (HERM) lines[22]. In
this paper, the short-window spectrograms are utilized in the analysis.

###  2.3 Detection and Classification

The approach taken in the paper is as follows, which is similar to prior work,
such as [23]). The time domain signal is first converted to a frequency domain
signal via an STFT. The separate real and imaginary components of the
resulting spectrogram are then used as input to the ML algorithms.

The regime of interest in practice is the low-SNR regime which is what is
studied in detail in this paper.

##  3 Simulation Setup

###  3.1 Data generation

We use a short-window Short Time Fourier Transform (STFT) with the two sided
spectra and a Hamming window with a window size of 16 samples and window
overlap of 8. The sample rate is set to be the same as the radar PRF: 10kHz
and the sampling time is 0.2s, sample length is therefore 2000.

The SNR values of -5 dB to -20 dB, in increments of 5 dB, are used for the
detector models. And for the classifier models, SNR values used range between
20 dB to -5 dB, in increments of 5 dB. We did not explore SNR values lower
than -5 dB for the classifier models, as the F1 scores deteriorate rapidly at
even lower SNR levels. This can be attributed to the inherent difficulty of
drone classification compared to drone detection.

###  3.2 Neural Network Architecture

####  3.2.1 CNN

Figure 7: Architecture of the CNN Classifier for Drone Type Classification.
CNN classifier model comprises a convolutional portion followed by a linear
portion. Input STFT spectrograms undergo convolutional layers with instance
normalization, Leaky ReLU activation, and periodic Max Pooling to extract
relevant features. Extracted features are flattened and passed through fully
connected layers with Leaky ReLU activation. The final layer produces class
probabilities for input radar signals, enabling classification among five
drone types.

The classical Convolutional Neural Network (CNN) model, illustrated in Fig.7,
is a two-part architecture consisting of a convolutional portion and a linear
portion. The input data, STFT spectrograms, undergo a series of convolutional
layers, each followed by instance normalization, Leaky Rectified Linear Unit
(Leaky ReLU) activation, and periodic Max Pooling. This process extracts
relevant features from the spectrograms. The extracted features are then
flattened and passed through three fully connected layers, each with Leaky
ReLU activation. The final layer produces the class probabilities for the
input radar signals.

To work as a detector and perform binary classification, the final fully
connected layer is designed to have two output channels. The model containing
this architecture will be referred to as the CNN detector.

To perform classification among the five drone types, the final fully
connected layer is modified to have five output channels. The model containing
this architecture will be referred to as the CNN classifier.

####  3.2.2 HQNN

Figure 8: Architecture of the HQNN detector model. STFT spectrograms undergo
convolutional processing for feature extraction and dimensionality reduction.
Processed features pass through classical fully connected layers, then split
into four tensors for quantum layer processing implemented as VQCs. After
quantum processing, tensors are concatenated and refined through a final
classical fully connected layer for binary classification. Max pooling and
dropout regularization reduce dimensionality and overfitting.

The architecture of the hybrid quantum neural network (HQNN), depicted in Fig.
8, employs a hybrid approach that integrates classical convolutional layers
with a blend of classical fully connected layers and parallel quantum dense
layers. Each quantum layer is implemented as a VQC.

The input to the network, STFT spectrograms, is processed through a sequence
of convolutional layers, instance normalization layers, dropout layers, and
max pooling layers. This initial convolutional processing extracts and
enhances relevant features from the spectrogram data while also reducing the
dimensionality.

The data are then flattened and passed through fully connected layers with
Leaky ReLU activation. The processed spectrogram features are then split into
four separate tensors, each of which is fed into a separate quantum layer.
These quantum layers introduce a non-linear quantum based transformation to
the features, potentially capturing more abstract and complex relationships
between them. Following the quantum circuit layers, the four processed feature
tensors are concatenated and passed through a final fully connected layer with
Leaky ReLU activation. These layers further refine and classify the extracted
features, leading to the final classification outputs. Max pooling layers are
used to reduce the dimensionality of the data. Dropout regularization is
applied after the max pooling operations to mitigate over-fitting.

The quantum part of the HQNN consists of $n$ parallel quantum layers [15].
Each layer is structured as a VQC that encompasses three main components:
embedding, variational gates, and measurement. The embedding is done using
“angle embedding” method. It encodes classical features into the rotation
angles of qubits. If the input data for each of the quantum layers is given by
the input vector
${x}=(\varphi_{1},\varphi_{2},\ldots,\varphi_{l})\in\mathbb{R}^{l}$, where
$l=m/n$ and $m$ represents the features from the previous classical fully
connected layer, then the encoding operation is given by
$|\psi\rangle=\text{R}_{x}(x)|\psi_{0}\rangle$ where
$|\psi_{0}\rangle=|0\rangle^{\otimes q}$ where $\text{R}_{x}$ is the rotation
gate applied around the X-axis of the Bloch sphere. It rotates each qubit,
which is in the ground state, by an angle proportional to the corresponding
value in the input vector. The variational part consists of single-qubit
rotations on each qubit with a trainable parameter, followed by a closed chain
of CNOT operations. The rotations modify the encoded input data based on the
variational parameters while the CNOT gates create entanglement among the
qubits. The depth of the variational component $i$ defines the iterations of
rotations and CNOT operations within each quantum layer. Variational
parameters vary for each repetition $i$ and for individual $c$ quantum
circuits. And the total number of weights in the quantum part of the HQNN is
given by $l\cdot 3i\cdot c$. The measurement part consists of the measurement
performed on the Pauli basis matrices. The measurement operation results in
$v^{(j)}=\langle
0|R_{x}(\phi_{j})^{\dagger}U(\theta)^{\dagger}Y_{j}U(\theta)R_{x}(\phi_{j})\left|0\right\rangle$
where, $Y_{j}$ is the Pauli Y matrix for the $j^{th}$ qubit. $R_{x}(\phi_{j})$
and $U(\theta)$ are the embedding operation and the operation by the trainable
part of the VQC, respectively. And $\theta$ is a vector of trainable
parameters.

Following the measurement stage, each quantum layer produces a vector
$v\in\mathbb{R}^{l}$. The resulting vectors from all the quantum layers are
concatenated to form a new vector $\hat{v}\in\mathbb{R}^{m}$ which is used as
the input data for the subsequent classical fully-connected layer.

To work as a detector and perform binary classification, the final fully
connected layer is designed to have two output channels. The model containing
this architecture will be referred to as the HQNN detector.

To perform classification among the five drone types, the final fully
connected layer is modified to have five output channels. The model containing
this architecture will be referred to as the HQNN classifier.

###  3.3 Model Training

HQNN detector and the CNN detector were trained on a dataset of 10,000 drone
images (equally among the five drone types) and 10,000 noise images. Model
evaluations were performed using a dataset of 5000 drone images (again,
equally among the five drone types) and 5000 noise images.

HQNN classifier and CNN classifier were trained on a dataset of 5000 drone
images (equally among the five drone types). Model evaluations were performed
using a dataset of 1000 drone images.

We train the CNN detector and the HQNN detector with SNR -5 db data, while we
train the CNN classifier and the HQNN classifier with SNR 5 db data.

Model training runs were executed for at least 50 epochs and achieved a loss
threshold of less than $0.005$. All training was done using PyTorch, a Python
machine learning library, and Pennylane, a python library for quantum machine
learning

##  4 Results and Discussion

###  4.1 Detection Models

SNR | CNN Detector F1 score | HQNN Detector F1 score
---|---|---
-5 dB | $0.963\pm 0.0015275$ | $0.960\pm 0.003464$
-10 dB | $0.917\pm 0.0040414$ | $0.946\pm 0.010263$
-15 dB | $0.824\pm 0.0183394$ | $0.898\pm 0.032036$
-20 dB | $0.745\pm 0.0273922$ | $0.843\pm 0.043485$
Table 2: F1 scores of detection models for various SNR levels. Both CNN
detector and HQNN detector are trained with SNR -5 dB data. Standard deviation
calculated using three datasets.

We use the F1 score to evaluate trained models, as it accounts for both false
positives and false negatives, unlike pure accuracy. The F1 score versus
various SNR values of the testing data for the classical and hybrid detection
model is shown in Table 2 and plotted in Fig. 9. The standard deviation is
calculated using three test datasets.

At comparatively higher SNR levels, at -5 dB and -10 dB, both the CNN and HQNN
detectors exhibit high F1 scores, with the HQNN detector slightly
outperforming. As the SNR decreases, at -15 dB and -20 dB, there is a
noticeable decrease in the F1 scores for both detectors. In particular, the
HQNN detector consistently maintains a higher F1 score compared to the CNN
detector at these lower SNR levels.

Figure 9: F1 Score comparison of CNN and HQNN Classifiers trained with SNR 5
dB data. Standard deviation was calculated with three datasets. HQNN detector
consistently maintains a higher F1 score compared to the CNN detector at lower
SNR levels.

We also use the Receiver Operating Characteristic (ROC) plot to better
understand the trade-off between true positives and false positives. This ROC
curve is particularly relevant here, as it is standard in radar performance
analysis. In addition, we used the log-linear plot for the ROC plot, as is
traditional in the radar literature. The ROC plot for the CNN detector for SNR
-20 is shown in Fig. 10 and for the HQNN detector in Fig.11.

The noticeable difference between the CNN detector and HQNN detector for the
ROC curve can be clearly seen at SNR -20 dB, with the HQNN detector having a
better true positive rate value.

Figure 10: ROC plot for CNN detector for SNR -20, trained at SNR -5 dB data.
Figure 11: ROC plot for HQNN detector at SNR -20 dB, trained at SNR -5 dB
data.

We also produce a binary confusion matrix to evaluate the detector models. The
confusion matrices for the CNN detector are given in Fig. 12 and in Fig. 13
for the HQNN detector. The difference between the detector models becomes
evident when examining the confusion matrices across all SNR levels, with
particularly notable differences observed at lower SNR levels. At SNR -20 dB,
more drones are classified as noise in both models, although the HQNN detector
performs better. This is plausible because STFTs at very low SNR values are
increasingly indistinguishable from STFT of Gaussian white noise.

Figure 12: Confusion matrix of CNN detector trained at SNR -5 dB data. Figure
13: Confusion matrix of HQNN detector trained at SNR -5 dB data.

###  4.2 Classification Models

SNR | CNN Classifier F1 | HQNN Classifier F1
---|---|---
20 dB | $0.981\pm 0.002082$ | $0.984\pm 0.002646$
15 dB | $0.969\pm 0.002517$ | $0.975\pm 0.001155$
10 dB | $0.949\pm 0.001527$ | $0.959\pm 0.005686$
5 dB | $0.905\pm 0.001527$ | $0.922\pm 0.004509$
0 dB | $0.813\pm 0.002517$ | $0.843\pm 0.002646$
-5 dB | $0.654\pm 0.009712$ | $0.696\pm 0.002887$
Table 3: F1 scores of classification models. Both the CNN Classifier and the
HQNN Classifier are trained with the SNR 5 dB data set. Standard deviation
calculated using three data sets.

For classifer models, SNR values below -5 dB wasn’t investigated, as F1 scores
degrade rapidly at lower SNR levels, likely because of the greater challenge
of drone classification versus detection. The F1 score for various SNR values
of the testing data for the classical and hybrid classification models is
shown in Table 3 and plotted in Fig. 14. Standard deviation was calculated
using three test datasets. At higher SNR levels, both CNN and HQNN classifiers
demonstrate high F1 scores, with the HQNN classifier consistently performing
slightly better. As the SNR decreases, there is a noticeable decrease in the
F1 scores for both classifiers. However, even at lower SNR levels, the HQNN
classifier tends to maintain a higher F1 score compared to the CNN classifier.

Figure 14: F1 Score against SNR comparison of CNN and HQNN Classifiers trained
with SNR 5 dB data. Standard deviation was calculated with three datasets.

We use Multi-class ROC plots to better gain insight into the performance of
the CNN classifier and HQNN classifier model. ROC plots for CNN classifier
trained with SNR 5 dB data are given in Fig. 15 and for HQNN classifier in
Fig. 16

The ROC curves differ with different drone types for both the CNN and HQNN
classifier models. The ROC curve for DJI Matrice 300 RTK for the HQNN
classifier is considerably worse than for the other drone types and the
corresponding performance in the CNN classifier. This could be attributed to
DJI Matrice 300 RTK having a very dense micro-Doppler signature (HERM lines)
[19]. At low SNR levels, for example at SNR -5 dB, the spectrograms of DJI
Matrice 300 RTK (See Fig. 4) could exhibit augmentation to a degree where it
starts to resemble the STFTs of Gaussian white noise (see Fig. 6), which has
no distinct signal. This is in contrast to the STFT of Parrot Disco at SNR -5
dB in Fig. 3 where the drone signal is more discernible. However, the ROC
plots for HQNN classifier do perform noticeably better than CNN classifier
across all SNR levels for at least some drone types for example, the Parrot
Disco.

Figure 15: ROC plots for CNN classifier trained with SNR 5 dB data  Figure 16:
ROC plots for HQNN classifier trained with SNR 5 dB data

We use multi-class confusion matrix to better gain insight into the
performance of the classifier models. Confusion matrices for CNN classifier
trained at SNR 5 dB data are shown at Fig. 17 and the confusion matrices for
HQNN classifier trained at SNR 5 dB data are shown at Fig. 18 The difference
between CNN classifier and HQNN classifier can be especially seen for SNR -5
dB. Both models seem to confuse other drone types for DJI Matrice 300 RTK more
often.

Figure 17: Confusion matrix of CNN classifier trained at SNR 5 dB data. The
labels correspond to the following drone models: 0 for DJI Matrice 300 RTK, 1
for DJI Mavic Air 2, 2 for DJI Mavic Mini, 3 for DJI Phantom 4, and 4 for
Parrot Disco.  Figure 18: Confusion matrix of HQNN classifier trained at SNR 5
dB data. The labels correspond to the following drone models: 0 for DJI
Matrice 300 RTK, 1 for DJI Mavic Air 2, 2 for DJI Mavic Mini, 3 for DJI
Phantom 4, and 4 for Parrot Disco.

##  5 Conclusion and Future Work

This article investigates the use of HQNNs for the problem of drone detection
and classification. We also compare the performance of HQNN with a classical
CNN of comparable architecture. We observed remarkable detection accuracy,
even at extremely low SNR levels of -5, -10, -15, -20, with both classical and
hybrid models. At SNR -20 dB, CNN achieves an F1 score of 0.745 ± 0.0273922
for drone detection, while HQNN outperforms it with an F1 score of 0.843 ±
0.0434856. At SNR -5 dB, CNN achieves an F1 score of 0.654 ± 0.009712 for
drone classification, whereas HQNN achieves a higher F1 score of 0.696 ±
0.002887 for the same task. Overall, HQNN outperforms a comparable CNN,
particularly noticeable at lower SNR levels, highlighting its importance in
investigating environments characterized by extremely low SNR levels.

In future work, one could take advantage of the transfer learning process; the
models can be re-trained with real data that may have nuances or variations
that are not perfectly captured with the synthetic data. A pre-trained model
would allow us to achieve good performance even with limited real data, which
is especially advantageous since real data is scarce. Future work could also
investigate other possible radar parameters, and also study different neural
network architectures, including using a quanvolutional layer, to develop more
robust solutions.

This work is preliminary. For example, there is potential for achieving
significant detection accuracy even at SNR values lower than -20 dB, when
neural network models are trained at appropriate SNR data. A more complicated
CNN architecture might outperform HQNN. Nevertheless, it is quite interesting
to note that HQNN achieves an unmistakable gain in this problem, and thus
motivates the need for further exploration of quantum-inspired algorithms to
this important practical problem.

## Code Availability

A version of the code used for this work is available in:
https://github.com/AishSweety/hybrid-quantum-classical-Neural-Network-for-
radar-data

## Conflicts of Interest

The authors declare no conflict of interest.

## Acknowledgements

We thank Tim Mathams for the insightful and helpful discussions with signal
processing and coding.

## Abbreviations

  * QML , Quantum Machine Learning

  * HQNN , Hybrid Quantum Neural Network

  * STFT , Short-Time Fourier Transform

  * MM , Martin-Mulgrew

  * CNN , Convolutional Neural Network

  * HERM , HElicopter Rotation Modulation

  * STFT , Short-Time Fourier Transform

  * SNR , Signal-to-Noise Ratio

  * PRF , pulse repetition frequency

  * ROC , Receiver Operating Characteristic

  * ReLU , Rectified Linear Unit

  * VQC , Variational Quantum Circuit

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