The growth of operator entropy in operator growth

Zhong-Ying Fan1†

1† Department of Astrophysics, School of Physics and Material Science,

Guangzhou University, Guangzhou 510006, China

ABSTRACT

We study upper bounds on the growth of operator entropy $S_{K}$ in operator
growth. Using uncertainty relation, we first prove a dispersion bound on the
growth rate $|\partial_{t}S_{K}|\leq 2b_{1}\Delta S_{K}$, where $b_{1}$ is the
first Lanczos coefficient and $\Delta S_{K}$ is the variance of $S_{K}$.
However, for irreversible process, this bound generally turns out to be too
loose at long times. We further find a tighter bound in the long time limit
using a universal logarithmic relation between Krylov complexity and operator
entropy. The new bound describes the long time behavior of operator entropy
very well for physically interesting cases, such as chaotic systems and
integrable models.

Email<EMAIL_ADDRESS>,

###### Contents

  1. 1 Introduction
  2. 2 The recursion method and the Lanczos algorithm
  3. 3 Upper bound on the growth of A-complexity
    1. 3.1 Conditions for saturation of the dispersion bound
  4. 4 Bound on the growth of K-entropy
    1. 4.1 Saturation of the dispersion bound
    2. 4.2 Continuum limit analysis
    3. 4.3 A new bound on K-entropy in the long time limit
      1. 4.3.1 Numerical results
  5. 5 Conclusion

##  1 Introduction

The Krylov complexity ( or K-complexity) [1] and the operator entropy (or
K-entropy) [2] are introduced to describe the Heisenberg evolution of
operators $\mathcal{O}(t)=e^{iHt}\mathcal{O}_{0}e^{-iHt}$, where
$\mathcal{O}_{0}$ is an initial operator and $H$ is the Hamiltonian. In
particular, for complex systems, the initially simple operator
$\mathcal{O}_{0}$ will irreversibly grow into a complex one, with the size
blowing up exponentially. It is widely believed that a statistical description
should emerge in this process and some universal features may be captured by
K-complexity and K-entropy.

The two notions have attracted a lot of attentions in literature [3, 4, 5, 6,
7, 8, 9, 10, 11, 12]. In particular, it was established in [10] that for
general irreversible process, the two quantities enjoy a universal logarithmic
relation to leading order at long times: $S_{K}=\eta\,\mathrm{ln}{K}+\cdots$,
where $0<\eta\leq 1$ and $\eta=1$ corresponds to chaotic systems111By chaotic
systems, we mean the Lanczos coefficients grows linearly asymptotically
$b_{n}\rightarrow\alpha n+\gamma$, except for $1\mathrm{d}$ case, in which
there is a logarithmic correction $b_{n}\rightarrow\alpha n/\mathrm{ln}{n}$.
However, the authors in [6] show that for free field theories at finite
temperatures, which do not probe chaos dynamically, the Lanczos coefficients
grow linearly asymptotically as well. This raises a question whether
asymptotic behavior of Lanczos coefficients characterizes dynamical properties
of Hamiltonian systems appropriately. Nonetheless, this does not influence our
discussions and results.. On the other hand, it was proved in [11] that the
growth rate of K-complexity in operator growth is upper bounded as
$|\partial_{t}K|\leq 2b_{1}\Delta K$, where $b_{1}$ is the first Lanczos
coefficient and $\Delta K$ is the variance of K-complexity. This is referred
to as dispersion bound [11]. This inspires us to search a similar bound for
K-entropy and examine the influence of the above logarithmic relation to the
growth of K-entropy.

The paper is organized as follows. In section 2, we briefly review the
recursion method and the Lanczos algorithm. In section 3, we prove a upper
bound on the growth of general A-complexity defined in the Kryolv space. We
show that saturation of the bound demands that A-complexity operator is
linearly related to K-complexity operator. In section 4, we study the upper
bound on the growth of K-entropy. Using continuum limit, we establishes that
the dispersion bound of K-entropy is rarely saturated. Instead, for
irreversible process, the logarithmic relation between K-complexity and
K-entropy implies a tighter bound in the long time limit. We test the new
bound using a number of numerical examples. We conclude in section 5.

##  2 The recursion method and the Lanczos algorithm

For a lattice system with Hamiltonian $H$, the initial operator
$\mathcal{O}_{0}$ evolves in time according to

$\mathcal{O}(t)=e^{iHt}\mathcal{O}_{0}e^{-iHt}=\sum_{n=0}{\frac{(it)^{n}}{n!}}\tilde{\mathcal{O}}_{n}\,,$
(1)

where $\tilde{\mathcal{O}}_{n}$ stands for the nested commutators:
$\tilde{\mathcal{O}}_{1}=[H,\mathcal{O}_{0}]\,,\tilde{\mathcal{O}}_{2}=[H,\tilde{\mathcal{O}}_{1}]\,,\cdots\,,\tilde{\mathcal{O}}_{n}=[H,\tilde{\mathcal{O}}_{n-1}]\,.$
However, sometimes evaluation of these commutators is of great difficulty. It
is helpful to take the operator as a wave function, which evolves under the
Liouvillian $\mathcal{L}\equiv[H\,,\cdot]$. One has

$|\mathcal{O}(t))=e^{i\mathcal{L}t}|\mathcal{O}_{0})=\sum_{n=0}{\frac{(it)^{n}}{n!}}|\tilde{\mathcal{O}}_{n})\,,$
(2)

where $|\tilde{\mathcal{O}}_{n})=\mathcal{L}^{n}|\mathcal{O}_{0}).$ To
proceed, one needs introduce an inner product in the operator Hilbert space
satisfying

$(A|A)\geq 0\,,\quad(A|B)=(B|A)^{*}\,,\quad(A|aB)=a(A|B)\,,$ (3)

where $a$ is a complex number. In addition, the Liouvillian should be
hermitian under the inner product $(A|\mathcal{L}B)=(\mathcal{L}A|B)$. It is
clear that the choice of inner product greatly influences the outcome of the
recursion method, such as the behavior of Lanczos coefficient $b_{n}$
introduced below. However, for our purpose in this paper, we do not need to
specify a particular choice of inner product ( when we discuss certain
Hamiltonian systems, the inner product generally has already been specified in
literature but we will not mention it explicitly. The readers should not be
confused). The interested readers are referred to [13] for more details about
choice of inner product.

The physical information about operator growth is essentially encoded in the
auto-correlation function

$C(t):=(\mathcal{O}_{0}|\mathcal{O}(t))=(\mathcal{O}_{0}|e^{i\mathcal{L}t}|\mathcal{O}_{0})\,.$
(4)

In general, the original operator basis $\\{|\tilde{\mathcal{O}}_{n})\\}$ is
not orthogonal. To study the operator dynamics in an orthonormal basis, one
adopts the Gram-Schmidt scheme, starting with a normalized vector
$|\mathcal{O}_{0})$. The first vector is given by
$|\mathcal{O}_{1}):=b_{1}^{-1}\mathcal{L}|\mathcal{O}_{0})$, where
$b_{1}:=(\mathcal{O}_{0}\mathcal{L}|\mathcal{L}\mathcal{O}_{0})^{1/2}$. For
the $n-$th vector, one has inductively

$\displaystyle|A_{n}):=\mathcal{L}|\mathcal{O}_{n-1})-b_{n-1}|\mathcal{O}_{n-2})\,,$
$\displaystyle|\mathcal{O}_{n}):=b_{n}^{-1}|A_{n})\,,\quad
b_{n}:=(A_{n}|A_{n})^{1/2}\,.$ (5)

The output of this procedure is a set of orthomornal vectors
$\\{|\mathcal{O}_{n})\\}$, referred to as Krylov basis and a sequence of
positive numbers $\\{b_{n}\\}$, referred to as Lanczos coefficients. These
coefficients have units of energy and can be used to measure time in the
Heisenberg evolution of operators.

In Krylov basis, evolution of the operator $\mathcal{O}(t)$ can be formally
written as

$|\mathcal{O}(t)):=\sum_{n=0}i^{n}\varphi_{n}(t)|\mathcal{O}_{n})\,,$ (6)

where $\varphi_{n}(t):=i^{-n}(\mathcal{O}_{n}|\mathcal{O}(t))$ is a discrete
set of (real) wave functions and $p_{n}\equiv\varphi_{n}^{2}$ can be
interpreted as probabilities. One has the normalization

$\sum_{n=0}^{\infty}\varphi^{2}_{n}(t)=1\,.$ (7)

The Heisenberg evolution of $\mathcal{O}(t)$ gives rise to a discrete set of
equations

$\partial_{t}\varphi_{n}=b_{n}\varphi_{n-1}-b_{n+1}\varphi_{n+1}\,,$ (8)

subject to the boundary condition $\varphi_{n}(0)=\delta_{n0}$ and
$b_{0}=0=\varphi_{-1}(t)$ by convention. This uniquely determines the wave
functions $\varphi_{n}(t)$ for a given set of Lanczos coefficients. Since the
auto-correlation function is simply $C(t)=\varphi_{0}(t)$, it is immediately
seen that physical information encoded in $C(t)$ can be equivalently extracted
from the Lanczos coefficients, though there is not a simple transformation
between them.

The Krylov complexity (K-complexity) and the Krylov entropy (K-entropy) are
defined as

$\displaystyle K$ $\displaystyle=$
$\displaystyle(\mathcal{O}(t)|\mathcal{K}|\mathcal{O}(t))\,,\quad\mathcal{K}=\sum_{n=0}n|\mathcal{O}_{n})(\mathcal{O}_{n}|\,,$
(9) $\displaystyle S_{K}$ $\displaystyle=$
$\displaystyle(\mathcal{O}(t)|\mathcal{S}|\mathcal{O}(t))\,,\quad\mathcal{S}=\sum_{n=0}\mathcal{S}_{n}\,|\mathcal{O}_{n})(\mathcal{O}_{n}|\,,$
(10)

where $\mathcal{S}_{n}\equiv-\,\mathrm{ln}{\varphi_{n}^{2}}$. Generalisation
of K-complexity is also interesting, for example the K-complexity of degree
$i$ is defined as

$K_{i}=(\mathcal{O}(t)|\mathcal{K}_{(i)}|\mathcal{O}(t))\,,\quad\mathcal{K}_{(i)}=\sum_{n=0}n^{i}|\mathcal{O}_{n})(\mathcal{O}_{n}|\,.$
(11)

These quantities roughly play the same role as K-complexity in characterizing
operator growth. However, except for simplicity, the definition of
K-complexity is special in the sense that it is deeply connected to the
complexity algebra, which characterizes the symmetry of Heisenberg evolution
[11]. We will study this in the next section. Here we just remind the readers
that if and only if the complexity algebra is closed, the operator wave
functions can be exactly solved and the dispersion bound on the growth rate of
K-complexity is saturated [11].

##  3 Upper bound on the growth of A-complexity

In [11], the authors first proved an upper bound on the growth rate of
K-complexity, referred to as dispersion bound. It was shown that saturation of
the bound is equivalent to closure of the complexity algebra, which is spanned
by
$\\{\mathcal{L}_{+}\,,\mathcal{L}_{-}\,,\mathcal{B}=[\mathcal{L}_{-}\,,\mathcal{L}_{+}]\\}$,
where

$\displaystyle\mathcal{L}_{+}=\sum_{n=0}b_{n+1}\,|\mathcal{O}_{n+1})(\mathcal{O}_{n}|\,,$
$\displaystyle\mathcal{L}_{-}=\sum_{n=0}b_{n+1}\,|\mathcal{O}_{n})(\mathcal{O}_{n+1}|\,.$
(12)

Notice the Liouvillian $\mathcal{L}=\mathcal{L}_{+}+\mathcal{L}_{-}$. In fact,
only when complexity algebra is closed, the operator wave functions
$\varphi_{n}$ can be solved exactly [3] and hence these cases provide
interesting examples to study operator growth. In this section, we would like
to extend the result [11] to general A-complexity defined under the Krylov
basis

$A(t)\equiv(\mathcal{O}(t)|\mathcal{A}|\mathcal{O}(t))\,,\quad\mathcal{A}=\sum_{n=0}\mathcal{A}_{n}|\mathcal{O}_{n})(\mathcal{O}_{n}|\,,$
(13)

where $\mathcal{A}$ is a self-adjoint superoperator. We demand that
$\mathcal{A}_{n}$ is either a polynomial of $n$, independent of time or
$\mathcal{A}_{n}=\mathcal{S}_{n}$ when it depends on time. Clearly
K-complexity and K-entropy are included in our notions of A-complexity. In the
following, we will generalise the dispersion bound of K-complexity to this
entire class of complexity.

First, by straightforward calculations, we deduce

$\displaystyle(\mathcal{O}(t)|\mathcal{A}\mathcal{L}|\mathcal{O}(t))=\sum_{n=0}-i\mathcal{A}_{n}\varphi_{n}\dot{\varphi}_{n}\,,$
$\displaystyle(\mathcal{O}(t)|\mathcal{L}\mathcal{A}|\mathcal{O}(t))=\sum_{n=0}i\mathcal{A}_{n}\varphi_{n}\dot{\varphi}_{n}\,,$
(14)

where a dot denotes the derivative with respect to the canonical time
coordinate. This gives

$(\mathcal{O}(t)|[\mathcal{A}\,,\mathcal{L}]|\mathcal{O}(t))=-i\sum_{n=0}\mathcal{A}_{n}\dot{p}_{n}\,.$
(15)

On the other hand, according to definition, the growth rate of A-complexity is
given by

$\partial_{t}A(t)=\sum_{n=0}\Big{[}\mathcal{A}_{n}+p_{n}\partial_{p_{n}}\mathcal{A}_{n}\Big{]}\dot{p}_{n}\,.$
(16)

If $\mathcal{A}_{n}$ is independent of time, the second term in the square
bracket vanishes whereas if $\mathcal{A}_{n}=\mathcal{S}_{n}$, this term still
does not contribute because of $\sum_{n=0}\dot{p}_{n}=0$. In both cases, we
arrive at

$-i\partial_{t}A=(\mathcal{O}(t)|[\mathcal{A}\,,\mathcal{L}]|\mathcal{O}(t))\,.$
(17)

In fact, the result can be derived more directly by taking the operator space
as an ordinary Hilbert space, which evolves under the “Hamiltonian”
$\mathcal{L}$. Using the density operator
$\rho\equiv|\mathcal{O}(t))(\mathcal{O}(t)|$ and
$A=\mathrm{Tr}(\rho\mathcal{A})$, one finds

$-i\partial_{t}A=-i\mathrm{Tr}(\dot{\rho}\mathcal{A})=\mathrm{Tr}([\mathcal{L}\,,\rho]\mathcal{A})=(\mathcal{O}(t)|[\mathcal{A}\,,\mathcal{L}]|\mathcal{O}(t))\,,$
(18)

where in the second equality, we have adopted the Liouville equation
$\dot{\rho}=i[\mathcal{L}\,,\rho]$.

With this result in hand and adopting the Robertson uncertainty relation
$2\Delta A\Delta\mathcal{L}\geq|\langle[\mathcal{A}\,,\mathcal{L}]\rangle|$,
we obtain

$|\partial_{t}A|\leq 2\Delta A\Delta\mathcal{L}=2b_{1}\Delta\mathcal{A}\,,$
(19)

where $\Delta
A=\sqrt{\langle\mathcal{A}^{2}\rangle-\langle\mathcal{A}\rangle^{2}}$ stands
for the dispersion of $\mathcal{A}$ with respect to some state $|A)$ and
$\Delta\mathcal{L}=b_{1}$. Notice that if the Krylov space is infinite
dimensional, it is necessary that the state $|A)$ is contained in the
intersection of the domains of $\mathcal{A}\mathcal{L}$ and
$\mathcal{L}\mathcal{A}$, otherwise the bound may not hold [14]. To avoid
confusion with the uncertainty relation with observables, the bound is
referred to as dispersion bound in [11] and we shall follow this convention.

Several comments are in order:

$\bullet$ The bound is valid to any kind of A-complexity, including
K-complexity and K-entropy. These two cases will be studied carefully later.

$\bullet$ The initial time $t=0$ is an extremal point, at which the bound is
always saturated for any A-complexity.

$\bullet$ A geometrical interpretation was given for saturation of the bound
for K-complexity [11]: The projective Krylov space can be identified with all
rank one orthogonal projections on the Krylov space. The bound is saturated if
and only if the corresponding curve in the Krylov space moves along the
gradient of Krylov complexity: meaning the dynamics is directed to the
direction that maximizes the local growth of K-complexity. This interpretation
can be generalised to any kind of A-complexity straightforwardly.

$\bullet$ There are several cases, in which the complexity algebra is closed.
It was proved [11] that only in these cases the dispersion bound for
K-complexity is saturated. Later, we will show that for general A-complexity,
saturation of the dispersion bound is possible if and only if $\mathcal{A}$ is
linearly related to $\mathcal{K}$.

$\bullet$ One can not obtain a tighter bound by using generalised uncertainty
relation because of $\langle\\{\mathcal{A}\,,\mathcal{L}\\}\rangle=0$, see
(3). However, this does not exclude the possibility that for certain cases
(such as irreversible process), the growth rate of A-complexity could be
tighter bounded at some time regimes. We will study this in detail for
K-entropy in the next section.

###  3.1 Conditions for saturation of the dispersion bound

Apparently it is not clear whether the dispersion bound of A-complexity can be
saturated during unitary evolution of operators. However, since the
uncertainty relation is obtained by applying the Cauchy-Schwarz inequality to
the two vectors $(\mathcal{A}-A)|\mathcal{O}(t))$ and
$(\mathcal{L}-\langle\mathcal{L}\rangle)|\mathcal{O}(t))$, saturation of the
bound is equivalent to the two vectors are linearly dependent. In other words,
the vector
$e^{-i\mathcal{L}t}(\mathcal{A}-A)e^{i\mathcal{L}t}|\mathcal{O}_{0})$ is
linearly related to $|\mathcal{O}_{1})$ because of
$\langle\mathcal{L}\rangle=0$. To study this carefully, we adopt the Baker-
Campbell-Hausdorff formula and find

$e^{-i\mathcal{L}t}(\mathcal{A}-A)e^{i\mathcal{L}t}|\mathcal{O}_{0})=\Big{[}-A(t)+\sum_{n=0}{\frac{(-it)^{n}}{n!}}L^{n}\Big{]}|\mathcal{O}_{0})\,,$
(20)

where $L^{n}$ stands for the nested commutators:
$L^{0}=\mathcal{A}\,,L^{1}=[\mathcal{L}\,,\mathcal{A}]\,,L^{2}=\big{[}[\mathcal{L}\,,\mathcal{A}]\,,\mathcal{A}\big{]}\,,\cdots\,.$

To evaluate these commutators, we introduce matrix basises
$e_{m\,,n}=|\mathcal{O}_{m})(\mathcal{O}_{n}|$ and represent a matrix as
$M=\sum M_{m\,,n}e_{m\,,n}$. Here $e_{n\,,n}$ corresponds to the diagonal
elements. For later purpose, we refer the elements associated to the basis
$e_{n+k,n}$ (or $e_{n,n+k}$) as $k$-diagonals (or $-k$-diagonals) of
operators. One has

$e_{m\,,n}|\mathcal{O}_{k})=\delta_{nk}|\mathcal{O}_{m})\,,\quad
e_{m\,,p}\,e_{q\,,n}=\delta_{pq}e_{m\,,n}\,.$ (21)

We are ready to compute the nested commutators $L^{n}$:

$\bullet$ First, $L^{0}=\mathcal{A}$ and
$L^{0}|\mathcal{O}_{0})=\mathcal{A}_{0}|\mathcal{O}_{0})$. Similar terms
giving $|\mathcal{O}_{0})$ are widely contained in the nested commutators
$L^{n}$. In fact, if the upper bound is saturated, these terms will be
cancelled by $A(t)$ term in (20). It provides an alternate way to extract
A-complexity in this case, see [11] for K-complexity case.

$\bullet$ Second, straightforward calculation gives
$L^{1}=\sum_{n=0}b_{n+1}(\mathcal{A}_{n}-\mathcal{A}_{n+1})(e_{n+1\,,n}-e_{n\,,n+1})$
and
$L^{1}|\mathcal{O}_{0})=b_{1}(\mathcal{A}_{0}-\mathcal{A}_{1})|\mathcal{O}_{1})$.
This gives the desired term proportional to $|\mathcal{O}_{1})$. However,
evaluation of $L^{2}$ yields

$L^{2}=\sum_{n=0}g(n)b_{n+1}b_{n+2}(e_{n+2\,,n}+e_{n\,,n+2})+\cdots\,,$ (22)

where
$g(n)=\mathcal{A}_{n}-\mathcal{A}_{n+1}-(\mathcal{A}_{n+1}-\mathcal{A}_{n+2})$
and we have omitted the diagonal terms, which are irrelevant to our
discussions. One finds
$L^{2}|\mathcal{O}_{0})=g(0)b_{1}b_{2}|\mathcal{O}_{2})+\cdots$. Obviously
this term cannot be cancelled by the A-complexity term and hence must be set
to zero for saturation of the upper bound.

$\bullet$ As a matter of fact, the highest subdiagonal in $L^{n}$ is
$n$-diagonal (or $-n$-diagonal) so that
$L^{n}|\mathcal{O}_{0})=(L^{n})_{n,0}|\mathcal{O}_{n})+\cdots$. Saturation of
the upper bound demands that all these terms vanish. It follows that the
$(n+2)$-diagonals of $L^{n+2}$ can be deduced from the $(n+1)$-diagonals of
$L^{n+1}$:

$L(n\,,k)=b_{n+k+2}\,L(n-1\,,k)-b_{k+1}\,L(n-1\,,k+1)\,,$ (23)

where $n\geq 1$ and $L(n\,,k)\equiv\big{(}L^{n+2}\big{)}_{n+k+2\,,k}$.
Mathematically for above recurrence relations, together with the initial
condition $L(0\,,k)=g(k)b_{k+1}b_{k+2}$, one has [11]

$L(n\,,k)=\prod_{j=k+1}^{n+k+2}b_{j}\sum_{q=0}^{n}(-)^{q}\binom{n}{k}g(k+q)\,.$
(24)

This implies that $g(k)=0$ for $k\geq 1$ to guarantee all $n$-diagonals of
$L^{n}$ vanishes. Hence, $\mathcal{A}_{n}$ must be a linear function of $n$:
$\mathcal{A}_{n}=\alpha n+\beta$, where $\alpha\,,\beta$ are constants
independent of $n$ (but may depend on time). In other words, saturation of the
upper bound for A-complexity is possible when A-complexity operator
$\mathcal{A}$ is linearly related to the K-complexity operator
$\mathcal{A}=\alpha\mathcal{K}+\beta$. Since the bound for K-complexity is
saturated if and only if the complexity algebra is closed [11], it is a little
surprising that the algebra influences the unitary evolution of operators more
widely than expected: it constrains the growth rate of any kind of
A-complexity.

##  4 Bound on the growth of K-entropy

###  4.1 Saturation of the dispersion bound

We move to study K-entropy. According to previous discussions, saturation of
the dispersion bound on K-entropy is only possible when the complexity algebra
is closed and hence the bound on K-complexity is saturated. These are only a
few cases [3, 11]: the complexity algebra is $\mathrm{SL}(2\,,\mathbb{R})$
algebra (the SYK model), $\mathrm{SU}(2)$ algebra and Heisenberg-Weyl algebra.
Exploring these cases, we only find two examples which saturate the K-entropy
bound.

The first is the evolution of operator $\mathcal{O}_{0}=\sigma_{x}$ under a
single qubit (two-level) Hamiltonian $H=\omega\sigma_{z}$, where $\sigma_{i}$
stands for the $i$-th Pauli matrix. This case belongs to $j=1/2$ case of the
$\mathrm{SU}(2)$ algebra. One has $b_{1}=\omega\,,b_{n>1}=0\,,$ and
$\varphi_{0}=\cos(\omega t)\,,\varphi_{1}=\sin(\omega t)$. However, since the
Krylov dimension is only $D=2$, it turns out that the dispersion bound for any
A-complexity is trivially saturated222In addition to the K-entropy, consider
$A=\sum_{n}P(n)\varphi_{n}^{2}(t)$, where $P(n)$ is a polynomial of $n$,
independent of time. It is easy to see $A=P(1)\varphi_{1}^{2}(t)\,,\Delta
A=|P(1)\varphi_{0}(t)\varphi_{1}(t)|$ so that $|\partial_{t}A|=2b_{1}\Delta
A$.. For example, evaluation of K-entropy yields

$\displaystyle S_{K}=-\cos^{2}(\omega t)\,\mathrm{ln}\,{\cos^{2}(\omega
t)}-\sin^{2}(\omega t)\,\mathrm{ln}\,{\sin^{2}(\omega t)}\,,$
$\displaystyle\Delta S_{K}=2\sin(\omega t)\cos(\omega
t)\,\mathrm{ln}\,{\cot(\omega t)}\,.$ (25)

It is easy to see $|\partial_{t}S_{K}|=2b_{1}\Delta S_{K}$.

The second example belongs to a special case of SYK model. The Lanczos
coefficient is proportional to $n$: $b_{n}=\omega n$, with the wave functions
given by $\varphi_{n}(t)=\tanh^{n}{(\omega t)}/\cosh{(\omega t)}$. It is easy
to see that for this case the K-entropy operator is linearly related to
K-complexity operator. Explicitly, the K-entropy and the variance are given by

$\displaystyle S_{K}=\cosh^{2}(\omega t)\,\mathrm{ln}\,{\cosh^{2}(\omega
t)}-\sinh^{2}(\omega t)\,\mathrm{ln}\,{\sinh^{2}(\omega t)}\,,$
$\displaystyle\Delta S_{K}=2\sinh(\omega t)\cosh(\omega
t)\,\mathrm{ln}\,{\coth(\omega t)}\,.$ (26)

Again $|\partial_{t}S_{K}|=2b_{1}\Delta S_{K}$. It is interesting to observe
that by demanding reality and positivity, the above results for K-entropy (and
K-complexity) can be obtained by taking $\omega\rightarrow i\omega$ from the
single-qubit system. This is unexpected. However, the trick is not valid to
general A-complexity.

###  4.2 Continuum limit analysis

From above discussions, one may gain an intuition that saturation of the
dispersion bound for K-entropy in the unitary evolution of operators is much
harder than that for K-complexity. In fact, for the latter, the bound is
always asymptotically saturated at long times for chaotic systems333The
condition for saturation of the dispersion bound of K-complexity is
$b_{n+1}^{2}-b_{n}^{2}$ is linear in $n$ whereas for chaotic systems the
Lanczos coefficient always grows asymptotically linearly. Hence, the bound is
saturated asymptotically at long times for general chaotic systems. but this
never happens for K-entropy (or any other A-complexity). To search a deeper
understanding for this, we adopt the continuum limit to study the dispersion
bound for the two quantities.

It was known that for semi-infinite chains, the continuum limit is good at
capture the leading long time behaviors of K-complexity and K-entropy using
coarse grained wave functions [2]. Introducing a lattice cutoff $\epsilon$ and
defining a coordinate $x=\epsilon n$ and the velocity $v(x)=2\epsilon b_{n}$.
The interpolating wave function is defined as $\varphi(x\,,t)=\varphi_{n}(t)$.
The wave equation (8) becomes

$\partial_{t}\varphi(x\,,t)={\frac{1}{2\epsilon}}\Big{[}v(x)\varphi(x-\epsilon)-v(x+\epsilon)\varphi(x+\epsilon)\Big{]}\,.$
(27)

Expansion in powers of $\epsilon$, one finds to leading order

$\partial_{t}\varphi=-v(x)\partial_{x}\varphi-{\frac{1}{2}}\partial_{x}v(x)\varphi+O(\epsilon)\,.$
(28)

This is a chiral wave equation with a position-dependent velocity $v(x)$ and
mass ${\frac{1}{2}}\partial_{x}v(x)$. The equation is much simplified in a new
frame $y$ defined as $v(x)\partial_{x}=\partial_{y}$ and a rescaled wave
function

$\psi(y\,,t)=\sqrt{v(y)}\,\varphi(y\,,t)\,.$ (29)

One finds

$(\partial_{t}+\partial_{y})\psi(y\,,t)=0+O(\epsilon)\,.$ (30)

The general solution is given by

$\psi(y\,,t)=\psi_{i}(y-t)\,,$ (31)

where $\psi_{i}(y)=\psi(y\,,0)$ stands for the initial amplitude. This tells
us that the leading order wave function simply moves ballistically in the
evolution. This approximation derives the growth of K-complexity correctly but
for K-entropy, some higher order corrections should be included for certain
cases, for example the Lanczos coefficients have a bounded support [2].
However, for our purpose, the leading order result is already enough (we will
turn to numerical approach when the leading order analysis breaks down ).

Normalization condition reads

$1=\sum_{n}|\varphi_{n}(t)|^{2}={\frac{1}{\epsilon}}\int\mathrm{d}x\,\varphi^{2}(x\,,t)={\frac{1}{\epsilon}}\int\mathrm{d}y\,\psi^{2}(y\,,t)\,.$
(32)

Evaluation of the K-complexity and the K-entropy yields

$\displaystyle K(t)$ $\displaystyle=$
$\displaystyle\sum_{n}n|\varphi_{n}(t)|^{2}={\frac{1}{\epsilon}}\int\mathrm{d}y\,{\frac{x(y+t)}{\epsilon}}\,\,\psi_{i}^{2}(y)\,,$
(33) $\displaystyle S_{K}(t)$ $\displaystyle=$
$\displaystyle-\sum_{n}|\varphi_{n}(t)|^{2}\,\mathrm{ln}{|\varphi_{n}(t)|^{2}}$
(34) $\displaystyle=$
$\displaystyle{\frac{1}{\epsilon}}\int\mathrm{d}y\,\psi_{i}^{2}(y)\Big{[}\,\mathrm{ln}{v(y+t)}-\,\mathrm{ln}{\psi_{i}^{2}(y)}\Big{]}\,.$

Similarly, one has

$\displaystyle K_{2}(t)$ $\displaystyle=$
$\displaystyle\sum_{n}n^{2}|\varphi_{n}(t)|^{2}={\frac{1}{\epsilon}}\int\mathrm{d}y\,{\frac{x^{2}(y+t)}{\epsilon^{2}}}\,\,\psi_{i}^{2}(y)\,,$
(35) $\displaystyle S_{K_{2}}(t)$ $\displaystyle=$
$\displaystyle\sum_{n}|\varphi_{n}(t)|^{2}\,\mathrm{ln}^{2}{|\varphi_{n}(t)|^{2}}$
(36) $\displaystyle=$
$\displaystyle{\frac{1}{\epsilon}}\int\mathrm{d}y\,\psi_{i}^{2}(y)\Big{[}\,\mathrm{ln}{v(y+t)}-\mathrm{ln}{\psi_{i}^{2}(y)}\Big{]}^{2}\,.$

Using these results, once the transformation between the two frames is known,
we are able to extract the leading time dependence of the quantities
immediately.

Here we are particularly interested in evaluating the variance of K-complexity
and K-entropy. Using Taylor expansion, we deduce to leading order at long
times

$\Delta
K\simeq{\frac{x^{\prime}(t)}{\epsilon}}\sqrt{Y_{2}-Y_{1}^{2}}\sim\partial_{t}K/b_{1}\,,$
(37)

where $Y_{n}\equiv{\frac{1}{\epsilon}}\int dy\,y^{n}\,\psi^{2}_{i}(y)$ and we
have ignored higher order terms (these terms are in the same order of
$x^{\prime}(t)/\epsilon$ for chaotic systems but this does not influence our
discussions). This implies that while saturation of the dispersion bound
happens only for a few cases, the bound still captures the long time behavior
of K-complexity well. In other words, even if the dispersion bound is not
saturated asymptotically in operator growth, the K-complexity still grows
fast, with a rate of change close to the dispersion bound. This is also
supported by our numerical results. For example, in Fig.1, we show this
explicitly for integrable models $b_{n}=\alpha n^{\delta}$.

The variance of K-entropy can be analyzed in the same way. We find to leading
order

$\Delta
S_{K}={\frac{1}{\epsilon}}\int\mathrm{d}y\,\psi_{i}^{2}(y)\,\mathrm{ln}^{2}{\psi_{i}^{2}(y)}-{\frac{1}{\epsilon^{2}}}\Big{[}\int\mathrm{d}y\,\psi_{i}^{2}(y)\,\mathrm{ln}{\psi_{i}^{2}(y)}\Big{]}^{2}\,,$
(38)

which is of order unity. It implies that compared to K-complexity, the
dispersion bound of K-entropy approaches to a constant at long times for
general irreversible process. This is ensured by our numerical results as well
(the interested readers are referred to the next subsection for details). In
particular, for certain cases in which the leading order analysis breaks down,
we find the result is still valid. Hence, we may take (38) as a definite
result for irreversible process.

Figure 1: The ratio of the growth rate of K-complexity to the dispersion bound
in operator growth for integrable models $b_{n}=\alpha n^{\delta}$, where
$\delta=3/8$ (solid), $2/3$ (dashed) and $3/4$ (dotted). The horizontal line
stands for the cases in which the bound is saturated during the evolution. It
is immediately seen that even if for these models, the bound is not saturated
asymptotically, complexity still grows fast, with the rate of change in the
same order of the dispersion bound.

However, the growth rate of K-entropy generally decreases in a power law at
long times $\partial_{t}S_{K}\sim 1/t^{\gamma}$, where $0<\gamma\leq 1$ [10]
except in the scrambling regime of chaotic systems (where $\partial_{t}S_{K}$
is a constant). This illustrates technically why the dispersion bound of
K-entropy is too loose at long times444However, for chaotic systems,
$\partial_{t}S_{K}$ is a constant, in the same order of the entropy variance
$\partial_{t}S_{K}\sim 2b_{1}\Delta S_{K}$. Hence, the logic does not explain
why in this case, the dispersion bound of K-entropy is not asymptotically
saturated as K-complexity.. This is also in accordance with physical
expectations. At sufficiently long times, the distribution of wave functions
at an instant $t$ nearly arrives at the maximal entropy (allowed by the
operator dynamics) so that in the next time step $t\rightarrow t+\delta t$,
the increasing of entropy is highly suppressed ($\delta S_{K}\sim\delta K/K$
according to (39)), while the dimension of the operator space, effectively
characterized by K-complexity increases monotonically without any such
constraint (this is said in the regime $0<K<S$, for systems with $S$ extensive
degrees of freedoms).

The above analysis tells us that unlike K-complexity, the dispersion bound of
K-entropy is too loose to characterize the long time behavior of K-entropy for
irreversible process. This motivates us to search a tighter bound for the
entropy growth. In the next subsection, we will show that such a bound indeed
exists in the long time limit.

###  4.3 A new bound on K-entropy in the long time limit

It was established [10] that for irreversible process, there exists a
universal logarithmic relation between K-complexity and K-entropy to leading
order at long times:

$S_{K}=\eta\,\mathrm{ln}{K}+\cdots\,,$ (39)

where the coefficient $\eta$ is bounded as $0<\eta\leq 1$. The upper bound
$\eta=1$ is saturated by chaotic systems. In particular, it implies that long
time dynamics of fast scramblers is particularly simple. Given the mean value
of the distribution $D$, the wave functions to leading order are described by
a uniform distribution so that the entropy is maximal. One has
$S_{K}\simeq\log{D}$ and $K\simeq D/2$. Up to a factor of $2$, the
K-complexity exactly measures the effective dimension of the Krylov space at
an instant. It turns out that existence of the logarithemic relation bounds
the growth of K-entropy further in the long time limit.

Consider K-complexity at first. On one hand, one has the dispersion bound
$|\partial_{t}K|\leq 2b_{1}\Delta K$ whereas the logarithmic relation implies
at long times

$|\partial_{t}K|=K\,|\partial_{t}S_{K}|/\eta\leq 2b_{1}\,K\Delta
S_{K}/\eta\,,$ (40)

where in the first equality we have ignored the subleading order corrections.
Hence, strictly speaking the result is valid only in the long time limit. All
relevant results below should be understood in the same way. Now we have two
bounds for complexity growth. However, we have learnt from the continuum limit
analysis that complexity growth rate can be close to the dispersion bound in
the long time limit for irreversible process. Hence, it is hard to imagine
that (40) provides a tighter bound. Then consistency of the two bounds leads
us to propose

$t\rightarrow\infty\,,\quad|\partial_{t}K|\leq 2b_{1}\Delta K\leq
2b_{1}\,K\Delta S_{K}/\eta\,,$ (41)

or written more explicitly

$t\rightarrow\infty\,,\quad\eta\Delta K/K\leq\Delta S_{K}\,.$ (42)

The relation can also be understood as the constant $\eta$ is bounded in the
long time dynamics as

$\eta\leq\lim_{t\rightarrow\infty}\Delta S_{K}K/\Delta K\,.$ (43)

Return to the growth of K-entropy. If the above result is correct, it implies
a tighter bound on the K-entropy growth in the long time limit

$t\rightarrow\infty\,,\quad|\partial_{t}S_{K}|\leq 2b_{1}\,\eta\Delta K/K\leq
2b_{1}\Delta S_{K}\,.$ (44)

This is a main result of this paper. We will show that though this new bound
is valid only in the long time limit, it well describes the growth of
K-entropy in the long time tails of irreversible process.

Before moving to explicit examples, let us show why (42) should be correct.
First, consider fast scramblers. The K-complexity saturates the dispersion
bound in the long time limit such that

$\Delta K/K=\partial_{t}K/(2b_{1}K)=\partial_{t}S_{K}/2b_{1}\leq\Delta
S_{K}\,.$ (45)

This proves (42). For general cases, $\Delta K\,,\Delta S_{K}$ at long times
can be estimated using the continuum limit analysis. One has $\Delta
K/K\sim\partial_{t}K/b_{1}K\sim\partial_{t}S_{K}/b_{1}$, which generally
decreases (in a power law to leading order) at long times. However the
variance of K-entropy approaches to a constant $\Delta S_{K}\sim O(1)$.
Therefore, (42) continues hold beyond a critical time $t_{c}$ at which
$\eta\Delta K(t_{c})/K(t_{c})=\Delta S_{K}(t_{c})$. In fact, we obtain a
stronger result

$\eta\leq 1\leq\lim_{t\rightarrow\infty}\Delta S_{K}K/\Delta K\,.$ (46)

The relation passes a variety of numerical tests.

####  4.3.1 Numerical results

We would like to test the new bound of K-entropy proposed in (44) using a
variety of numerical examples. Notice that (44) is equivalent to (42) or (43)
so we shall focus on the former. Besides, it was shown in [10] that the
logarithmic relation (39) begins to emerge around the time scale $t_{*}$,
where $K(t_{*})\sim O(1)$. Here $t_{*}$ coincides with the scrambling time for
fast scramblers. It is interesting to see whether the result (44) can be
extended to finite times $t\geq t_{c}$ as well.

Consider SYK-like model at first. The Lanczos coefficient is given by
$b_{n}=\omega\sqrt{n(n-1+\xi)}$. The wave functions can be solved exactly as
[1]

$\varphi_{n}(t)=\sqrt{{\frac{(\xi)_{n}}{n!}}}\,{\frac{\tanh^{n}(\omega
t)}{\cosh^{\xi}{(\omega t)}}}\,,$ (47)

where $(\xi)_{n}=\xi(\xi+1)\cdots(\xi+n-1)$ is the Pochhammer symbol. For
$\xi=1$, the K-entropy and its variance have already been given by (4.1). It
turns out that in this case, the bound (44) is approached from below in the
long time limit: $\partial_{t}S_{K}=2b_{1}\Delta K/K=2b_{1}\Delta S_{K}$, as
shown in the left panel of Fig.2. Here it is worth emphasizing that this is
the only case we find in which the equality in (42) is taken. However,
generalisation of the relation (44) to finite times turns out to be incorrect.
Instead, since the dispersion bound is saturated during the evolution and
$t_{c}\rightarrow\infty$, one finds $\partial_{t}S_{K}=2b_{1}\Delta S_{K}\leq
2b_{1}\Delta K/K$.

Figure 2: Entropy growth for SYK model. Left panel: $\xi=1$. The bound
$\partial_{t}S_{K}=2b_{1}\Delta K/K$ is approached from below in the long time
limit and $\partial_{t}S_{K}=2b_{1}\Delta K/K=2b_{1}\Delta S_{K}$. However,
during the evolution, the dispersion bound is always saturated and
$\partial_{t}S_{K}=2b_{1}\Delta S_{K}\leq 2b_{1}\Delta K/K$ at finite (but
long) times. Right panel: $\xi=2$. The bound $\partial_{t}S_{K}=2b_{1}\Delta
K/K$ is again approached from below but the relation (44) can be extended to
finite times $t\geq t_{c}$.

The situation is much different for other $\xi$’s value. For example, the
$\xi=2$ case is shown in the right panel of Fig.2. The tighter bound proposed
in (44) is again approached from below but now $\partial_{t}S_{K}=2b_{1}\Delta
K/K<2b_{1}\Delta S_{K}$ in the long time limit. In particular, the relation
(44) can be extended to finite times $t\geq t_{c}$. As a matter of fact, we
study a number of chaotic models numerically and find these features are
generic.

We move to study integrable models which have $b_{n}=\alpha n^{\delta}$, where
$0<\delta<1$. In this case, the coefficient $\eta$ in the logarithmic relation
(39) is $\eta=\delta$ when $\delta\geq 1/2$ [10]. In Fig.3, we show the
entropy growth during the evolution for two examples555For Heisenberg-Weyl
case $\delta=1/2$, the wave functions can be solved exactly as
$\varphi_{n}(t)=e^{-\alpha^{2}t^{2}/2}{\frac{\alpha^{n}t^{n}}{\sqrt{n!}}}$. :
$\delta=1/2$ (left) and $\delta=2/3$ (right). It follows that for integrable
models, the growth rate of K-entropy generally decreases as
$\partial_{t}S_{K}\sim 1/t$ whereas $\Delta S_{K}$ still approaches to a
constant in the long time limit, consistent with our continuum limit analysis.
Furthermore, the entropy growth rate is very close to the tighter bound (44)
at long times. However, extension of relation (44) to finite times is not
always correct. For instance, for the Heisenberg-Weyl case, the bound
$\partial_{t}S_{K}=2b_{1}\eta\Delta K/K$ is approached from above so that the
entropy growth rate is also bounded from below $2b_{1}\eta\Delta
K/K\leq\partial_{t}S_{K}\leq 2b_{1}\Delta S_{K}$ at $t\geq t_{c}$ regime.

Finally, we would like to study the case in which the Lanczos coefficient has
a bounded support. In this case, the continuum limit analysis at leading order
is failed to capture the leading time dependence of K-entropy (but the result
for K-complexity is still valid) [2] . Hence, our result for $\Delta S_{K}$ in
sec.4.2 breaks down and the bound proposed in (44) looks problematic in this
case. However, our numerical results still support that $\Delta
S_{K}\rightarrow\mathrm{const}$ in the long time limit and the bound (44)
continues hold. For example, consider the simplest case $b_{n}=b$ which has
$\eta\simeq 0.729302$. The wave function is solved in terms of Bessel
functions

$\varphi_{n}(t)=J_{n}(2bt)+J_{n+2}(2bt)\,.$ (48)

The numerical results for entropy growth are shown in Fig.4. It is immediately
seen that up to subleading oscillations, $\Delta S_{K}$ approaches to a
constant while the entropy growth rate still decreases in a power law
$\partial_{t}S_{K}\sim b_{1}\Delta K/K\sim 1/t$.

Figure 3: Entropy growth for integrable model $b_{n}=\alpha n^{\delta}$. Left
panel: the Heisenberg-Weyl case $\delta=1/2$. The bound
$\partial_{t}S_{K}=2b_{1}\Delta K/K$ is approached from above and
$2b_{1}\eta\Delta K/K\leq\partial_{t}S_{K}<2b_{1}\Delta S_{K}$ beyond the
critical time scale $t_{c}$. Right panel: $\delta=2/3$. The bound
$\partial_{t}S_{K}=2b_{1}\Delta K/K$ is approached from below and the relation
(44) can be extended to finite times $t\geq t_{c}$.

Combining all the above results together, we conclude that during irreversible
operator growth, the K-entropy growth can be roughly classified into two
regimes: 1) the initial regime, in which the entropy growth rate nearly
saturates the dispersion bound $\partial_{t}S_{K}\simeq 2b_{1}\Delta S_{K}$;
2) the late time regime, in which the growth rate can be well described by the
new bound (44) $\partial_{t}S_{K}\sim 2b_{1}\eta\Delta K/K$. For physically
interesting cases, such as chaotic systems and integrable models, this
describes the growth of K-entropy very well during the Heisenberg evolution.

##  5 Conclusion

Figure 4: The K-entropy growth for the bounded support case $b_{n}=b$.

In this paper, we examine the upper bound on the growth of general
A-complexity defined in the Krylov space, including K-complexity and
K-entropy, during the Heisenberg evolution of operators. We first prove a
dispersion bound (19), generalising the result of K-complexity [11] to general
A-complexity. Our new contributions are: 1) We show that saturation of the
dispersion bound on A-complexity is possible if and only if the A-complexity
operator $\mathcal{A}$ is linearly related to the K-complexity operator
$\mathcal{K}$: $\mathcal{A}=\alpha\mathcal{K}+\beta$, where $\alpha\,,\beta$
are c-number constants (independent of $n$ but may depend on time). However,
since saturation of the bound for K-complexity is equivalent to closure of the
complexity algebra [11], this implies that the complexity algebra constrains
the growth of general A-complexity in a subtle way. 2) Though there are only a
few cases in which the complexity algebra is closed, the K-complexity grows
sufficiently fast at long times for general irreversible process, with the
rate of change close to the dispersion bound $\partial_{t}K\sim 2b_{1}\Delta
K$.

However, the situation for K-entropy is quite different. Except chaotic
systems, the entropy growth rate generally decreases in a power law to leading
order at long times $\partial_{t}S_{K}\sim 1/t^{\gamma}\,,0<\gamma\leq 1$
($\gamma=0$ for chaotic systems ) whereas the variance $\Delta S_{K}$
approaches to a constant in the long time limit. Thus generally the dispersion
bound on K-entropy is too loose to be saturated. However, it turns out that
the dispersion bound of K-complexity leads to a tighter bound on the growth of
K-entropy (44) in the long time limit because of a universal logarithmic
relation between K-complexity and K-entropy for irreversible process [10]. In
particular, for physically interesting cases such as chaotic systems and
integrable models, the K-entropy growth rate at long times is very close to
this new bound.

## Acknowledgments

Z.Y. Fan was supported in part by the National Natural Science Foundations of
China with Grant No. 11873025.

## References

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# The Prime Geodesic Theorem for the Picard Manifold

Ikuya Kaneko  The Division of Physics, Mathematics and Astronomy, California
Institute of Technology, 1200 East California Boulevard, Pasadena, CA 91125,
USA https://sites.google.com/view/ikuyakaneko/<EMAIL_ADDRESS>
(Date: August 28, 2024)

###### Abstract.

We prove the prime geodesic theorem for the Picard manifold whose error term
shrinks as the hybrid subconvex exponent $\theta$ for quadratic Dirichlet
$L$-functions over Gaussian integers decreases. Our rate of decay is faster
than that of Balkanova–Frolenkov (2022) and Kaneko (2022), and leads to the
exponent $\frac{202}{139}+\varepsilon$ in the prime geodesic theorem under the
generalised Lindelöf hypothesis $\theta=0$, improving upon the existing
conditional results. This provides further solid evidence towards the validity
of the conjectural exponent $1+\varepsilon$.

###### Key words and phrases:

Prime geodesic theorem, Picard manifold, character sums, subconvexity

###### 2020 Mathematics Subject Classification:

11L40 (primary); 11F72, 11M26 (secondary)

The author acknowledges the support of the Masason Foundation.

###### Contents

  1. 1 Introduction
  2. 2 Algebraic Preliminaries
  3. 3 Auxiliary Lemmata
  4. 4 Koyama’s Machinery
  5. 5 Proof of Theorem 1.1

##  1\. Introduction

###  1.1. Historical Prelude

The prime geodesic theorem asks for determining the asymptotic behaviour of
the counting function of primitive closed geodesics on hyperbolic manifolds.
If the underlying group is $\Gamma=\mathrm{PSL}_{2}(\mathbb{Z})$, then there
are a plethora of distinguished results in the literature; see [CG18, BBHM19,
Iwa84a, Iwa84b, LS95, Cai02, SY13, Kan24] and references therein.

In this paper, we study the prime geodesic theorem for arithmetic hyperbolic
$3$-manifolds where the upper-half plane $\mathfrak{h}^{3}$ is viewed as a
subset of Hamiltonian quaternions with vanishing fourth coordinate. It comes
naturally with a transitive action of $\mathrm{SL}_{2}(\mathbb{C})$ by linear
fractional transformations, and the stabiliser of any point is a conjugate of
$\mathrm{SU}_{2}(\mathbb{C})$. Hence, we identify

$\mathfrak{h}^{3}\cong\mathrm{SL}_{2}(\mathbb{C})/\mathrm{SU}_{2}(\mathbb{C})\cong\mathrm{Z}(\mathbb{C})\backslash\mathrm{GL}_{2}(\mathbb{C})/\mathrm{U}_{2}(\mathbb{C}).$

Given the basis $\\{1,i,j,k\\}$ of Hamiltonian quaternions, a point
$v\in\mathfrak{h}^{3}$ is denoted by $v=z+rj$ with $r>0$ and
$z=x+yi\in\mathbb{C}$; thus $\mathrm{Im}(v)\coloneqq r$,
$\mathrm{Im}(z)\coloneqq y$, and $\mathrm{Re}(z)\coloneqq x$. The space
$\mathfrak{h}^{3}$ is equipped with the hyperbolic metric
$r^{-1}\sqrt{dx^{2}+dy^{2}+dr^{2}}$ and the corresponding volume element
$d\mu(v)=r^{-3}dxdydr$. Then the quotient
$\mathrm{PSL}_{2}(\mathbb{Z}[i])\backslash\mathfrak{h}^{3}$ is termed the
Picard manifold. Henceforth, we write $\Gamma=\mathrm{PSL}_{2}(\mathbb{Z}[i])$
unless otherwise specified.

We now define the Chebyshev-like counting function by

$\Psi_{\Gamma}(X)\coloneqq\sum_{\mathrm{N}(P)\leq X}\Lambda_{\Gamma}(P),$

where the sum runs through all hyperbolic and loxodromic (with non-real trace)
conjugacy classes of $\Gamma$ of norm at most $X$, and
$\Lambda_{\Gamma}(P)=\log\mathrm{N}(P_{0})$ if $P_{0}$ is the primitive closed
geodesic underlying $P$, and $\Lambda_{\Gamma}(P)=0$ otherwise. Note that if
$\mathrm{N}(P)$ denotes the norm of a conjugacy class in question, then the
length of $P$ is equal to $\log\mathrm{N}(P)$. We refer to [EGM98, Section
5.7] for a thorough account of the terminology as well as the theoretical
background.

The first unconditional attempt towards an effective asymptotic for
$\Psi_{\Gamma}(X)$ traces back to the work of Sarnak [Sar83, Theorem 5.1],
which holds for any cofinite Kleinian group $\Gamma$:

$\mathcal{E}_{\Gamma}(X)\coloneqq\Psi_{\Gamma}(X)-\sum_{1<s_{j}\leq
2}\frac{X^{s_{j}}}{s_{j}}\ll_{\varepsilon}X^{\frac{5}{3}+\varepsilon},$ (1.1)

where the subtracted term emerges from the small eigenvalues
$\lambda_{j}=s_{j}(2-s_{j})=1+t_{j}^{2}<1$ of the hyperbolic Laplacian on
$\Gamma\backslash\mathfrak{h}^{3}$. If
$\Gamma=\mathrm{PSL}_{2}(\mathbb{Z}[i])$, then the existence of an explicit
form of the Kuznetsov formula due to Motohashi [Mot96, Mot97, Mot01] enables
an improvement over the exponent $\frac{5}{3}+\varepsilon$ in (1.1). In this
respect, Balkanova et al. [BCC+19, Theorem 1.1] proved the exponent
$\frac{13}{8}+\varepsilon$ via due adaptations of the machinery of Luo–Sarnak
[LS95] to the setting of $\mathbb{Q}(i)$. In emulation of previous work, Balog
et al. [BBCL22, Corollary 1.4] derived (via the Bykovskiĭ-type theorem in
three dimensions) an improved bound

$\mathcal{E}_{\Gamma}(X)\ll_{\varepsilon}X^{\frac{3}{2}+\frac{4\vartheta}{7}+\varepsilon},$

where $\vartheta$ is a subconvex exponent for quadratic Dirichlet
$L$-functions over Gaussian integers in the conductor aspect, namely

$L\left(\frac{1}{2}+it,\chi_{D}\right)\ll_{\varepsilon}(1+|t|)^{A}\mathrm{N}(D)^{\vartheta+\varepsilon}$
(1.2)

for all primitive quadratic characters $\chi_{D}$ over $\mathbb{Q}(i)$ with a
fundamental discriminant $D\in\mathbb{Z}[i]$. Note that the constant $A>0$ in
(1.2) is unspecified. The best unconditional exponent to date is given by
Balkanova–Frolenkov [BF22, Theorem 1.1] and the author [Kan22, Theorem 1.3]:

$\mathcal{E}_{\Gamma}(X)\ll_{\varepsilon}X^{\delta_{0}(\vartheta)+\varepsilon},\qquad\delta_{0}(\vartheta)\coloneqq\frac{3}{2}+\frac{32\vartheta^{2}+28\vartheta-1}{46+40\vartheta}.$
(1.3)

For reference, see the table below for a chronology of the known exponents and
the numerical values thereof (both in terms of fractions and decimals); we
substitute the best known result $\vartheta=\frac{1}{6}$ of Nelson [Nel20,
Theorem 1.1] as needed. Koyama [Koy01, Theorem 1.1] assumes in practice the
mean-Lindelöf hypothesis for Rankin–Selberg $L$-functions associated to
Hecke–Maaß cusp forms on $\Gamma\backslash\mathfrak{h}^{3}$, but for
expository brevity we here exploit [BF22, Theorem 1.3] to replace his
assumption with the generalised Lindelöf hypothesis $\vartheta=0$ and to keep
consistency in the table. We also mention to avoid confusion that the
definition of a subconvex exponent of Balkanova–Frolenkov [BF20, Equation
(3.24)] differs from ours.

Year | Author(s) | Exponent
---|---|---
$1983$ | Sarnak [Sar83] | $\frac{5}{3}=1.66666\cdots$
$2019$ | Balkanova et al. [BCC+19] | $\frac{13}{8}=1.625$
$2022$ | Balog et al. [BBCL22] | $\frac{67}{42}=1.59524\cdots$
$2022$ | Balkanova–Frolenkov [BF22] & Kaneko [Kan20, Kan22] | $\frac{376}{237}=1.58649\cdots$
$2001$* | Koyama [Koy01] | $\frac{11}{7}=1.57143\cdots$
$2020$* | Balkanova–Frolenkov [BF20] | $\frac{3}{2}=1.5$
$2022$* | Balog et al. [BBCL22] | $\frac{34}{23}=1.47826\cdots$
*An asterisked result is conditional on the generalised Lindelöf hypothesis $\vartheta=0$.

By [Kan22, Theorem 5.3], an additional assumption of substantial cancellations
in spectral exponential sums [Kan22, Conjecture 5.1] would lead to the
exponent $\frac{10}{7}=1.42857\cdots$, which remains the best known
conditional result in the literature, albeit being far away from the
conjectural exponent $1+\varepsilon$. This result is excluded from the above
table due to too strong the underlying assumptions. The subject of this paper
is to update the table with improved conditional results and to demonstrate
further solid evidence towards the exponent $1+\varepsilon$.

###  1.2. Statement of Results

Throughout the body of the paper, the letter $\Gamma$ stands for the Picard
group $\mathrm{PSL}_{2}(\mathbb{Z}[i])$. We interchangeably employ (as needed)
the Vinogradov asymptotic notation $\ll$ and $\gg$ and the big $O$ notation
$O(\cdot)$. Dependence on a parameter will always be denoted by a subscript.
For further details of the notation and conventions, see Section 1.4.

As a continuation of the prequel [Kan24], we establish the following
nontrivial bound.

######  Theorem 1.1.

For a fundamental discriminant $D$, let $\chi_{D}=(\frac{D}{\cdot})$ be the
primitive quadratic character modulo $(D)$ over $\mathbb{Q}(i)$, and suppose
that the hybrid subconvex bound111We use a different font $\theta$ for
distinction from the subconvex exponent $\vartheta$ in (1.2) in the conductor
aspect. In what follows, one may replace $\vartheta$ with $\theta$ as needed,
but not vice versa.

$L\left(\frac{1}{2}+it,\chi_{D}\right)\ll_{\varepsilon}(\mathrm{N}(D)(1+|t|))^{\theta+\varepsilon}$
(1.4)

holds for some $0\leq\theta<\frac{1}{4}$. Then we have for any $\varepsilon>0$
that

$\mathcal{E}_{\Gamma}(X)\ll_{\varepsilon}X^{\delta(\theta)+\varepsilon},$

where

$\delta(\theta)\coloneqq\frac{606+1558\theta+1516\theta^{2}+576\theta^{3}+64\theta^{4}}{417+871\theta+576\theta^{2}+80\theta^{3}}.$
(1.5)

$\frac{34}{23}$$\frac{202}{139}$0$\frac{1}{6}$$\frac{1}{4}$$\delta(\theta)$$\delta_{0}(\theta)$
Figure 1. A comparison of (1.5) and (1.3) as $\theta\in[0,\frac{1}{4})$
varies.

Figure 1 compares (up to $\varepsilon$) the quality of $\delta(\theta)$ and
$\delta_{0}(\theta)$ as $\theta\in[0,\frac{1}{4})$ varies, and visualises how
rapidly the former shrinks as $\theta$ approaches $0$, when compared to the
latter. In particular, immediate consequences of Theorem 1.1 include the
following bound.

######  Corollary 1.2.

Assume the generalised Lindelöf hypothesis $\theta=0$. Then we have for any
$\varepsilon>0$ that

$\mathcal{E}_{\Gamma}(X)\ll_{\varepsilon}X^{\frac{202}{139}+\varepsilon}.$

Because the numerical value of the exponent is
$\frac{202}{139}=1.45324\cdots$, Corollary 1.2 provides a $1.69$ % improvement
to the conditional exponent $\frac{34}{23}=1.47826\cdots$ as in the table
above. The importance of our results lies literally in conditional
improvements to the existing exponents, because the prime geodesic theorem for
the Picard manifold is still shrouded in mystery in the sense that a
legitimate heuristic for the conjectural exponent $1+\varepsilon$ is not known
to date, unlike in two dimensions [Iwa84a, Iwa84b, BFR22]. Hence, Corollary
1.2 further reinforces our belief that the exponent $1+\varepsilon$ is
reasonable, supporting substantial cancellations in the corresponding spectral
exponential sums. This conjecture, if true, should be optimal via the
$\Omega$-result of Nakasuji [Nak00, Nak01, Nak04]. We direct the reader to
[Kan22, Kan23, PR17] for an asymptotic formula for spectral exponential sums
in the spectral aspect with numerical experiments and speculations thereof.

######  Remark 1.3.

The work of Petrow–Young [PY20, PY23, You17] shows hybrid Weyl-strength
subconvex bounds for Dirichlet $L$-functions. An attempt to extend their
results to arbitrary number fields are made by Nelson [Nel20, Theorem 1.1],
where the archimedean aspect is out of consideration, thus we are unable to
specify the constant $A>0$ in (1.2). A generalisation in the hybrid aspect is
first considered by Balkanova–Frolenkov–Wu [BFW22, Theorem 1.1]. Nonetheless,
since it focuses on Hecke characters over totally real number fields, the
hybrid subconvex exponent $\theta=\frac{1}{6}$ that we desire is unavailable
at the current state of knowledge.

###  1.3. Technical Remarks

This section explicates an interpretation of Theorem 1.1 from a broad
perspective, with an emphasis on some ideas implicit in our proofs. The key
ingredients of the proof of Theorem 1.1 involve, among other manoeuvres,

  1. (1)

subconvex bounds for character sums over Gaussian integers;

  2. (2)

the stationary phase analysis in Koyama’s machinery [Koy01];

  3. (3)

a (systematic) worsening trick for spectral exponential sums;

  4. (4)

an interpolation with a result of Balkanova et al. [BCC+19].

In conjunction with the case of the modular surface, our proof relies on
subconvex bounds for quadratic character sums over Gaussian integers.
Nonetheless, the quality of the resulting bound (in particular the exponent of
$x$ in Lemma 3.3) becomes stronger. Loosely speaking, this phenomenon
exemplifies the shrinkage with respect to the degree of an algebraic number
field in the work of Hinz [Hin86] extending classical Burgess bounds. We
stress that our proof does not strictly utilise Burgess bounds, which explains
why we write here subconvex bounds.

Furthermore, we also strengthen the techniques of Koyama [Koy01] to take an
additional oscillatory factor into account. To this end, we incorporate a
refined version of the stationary phase analysis in Section 3.1. Combined with
a certain arithmetic expression of Kloosterman sums, the technology of Iwaniec
[Iwa84a], Cai [Cai02], and the author [Kan24] reduces our analysis to the
estimation of character sums over Gaussian integers, addressed in (1).

Nonetheless, these manipulations are followed by a bottleneck, namely there is
a possibility that the error term in the smooth explicit formula for
$\Psi_{\Gamma}(X)$ may dominate our final estimate for spectral exponential
sums. We eschew this problem via worsening the exponent of $X$ on purpose,
which makes the error term smaller and the other larger. This enables a
nontrivial optimisation of an auxiliary parameter $T$ in Section 5. Adapting
the methodology in [Kan24, Section 3], we consequently interpolate between our
estimate and a $3$-dimensional analogue of the Luo–Sarnak bound due to
Balkanova et al. [BCC+19], and employ the Bykovskiĭ-type theorem of Balog et
al. [BBCL22, Theorem 1.3]; cf. [Byk94, Kuz78, SY13] in two dimensions.

###  1.4. Notation and Conventions

All implicit constants throughout the paper are allowed to depend on
$\varepsilon>0$ that represents an arbitrarily small positive quantity, but
not necessarily the same at each instance. The Vinogradov symbol $A\ll B$ or
the big $O$ notation $A=O(B)$ means that $|A|\leq C|B|$ for some $C>0$. We
exploit the sign convention $t_{j}>0$ for spectral parameters as well as an
abuse of notation $\sum_{|t_{j}|\leq T}$ to indicate that the sum is
symmetrised by including the contributions of both $t_{j}$ and $-t_{j}$ in the
summand.

### Acknowledgements

The author thanks Dimitrios Chatzakos for helpful feedback.

##  2\. Algebraic Preliminaries

Recall that the ring of integers of $\mathbb{Q}(i)$ is denoted by
$\mathbb{Z}[i]$ and that the class number of $\mathbb{Q}(i)$ is one (hence
every ideal is principal). Let $\mathfrak{m}$ be a nonzero ideal of
$\mathbb{Z}[i]$. A Dirichlet character modulo $\mathfrak{m}$ is a group
homomorphism

$\chi_{\mathfrak{m}}:\mathrm{Cl}^{\mathfrak{m}}\to S^{1},$

where $\mathrm{Cl}^{\mathfrak{m}}$ denotes the narrow ray class group modulo
$\mathfrak{m}$, which is an algebraic alternative to
$(\mathbb{Z}/m\mathbb{Z})^{\times}$ where the base field is $\mathbb{Q}$ and
$\mathfrak{m}=m\mathbb{Z}$.

Throughout, we study quadratic characters associated to the Jacobi–Kronecker
symbol222The notation differs from the first display on [BBCL22, Page 1899] as
the numerator and denominator are interchanged. Nonetheless, by quadratic
reciprocity, this difference does not affect our ensuing computations.

$\chi_{D}(n)=\left(\frac{D}{n}\right),$

where $D$ and $n$ are nonzero elements in $\mathbb{Z}[i]$; see [IR90, Section
14.2]. It is a primitive Dirichlet character modulo the ideal
$\mathfrak{m}=(D)$. In conjunction with the rational case, summing over
nonzero ideals $n\subseteq\mathbb{Z}[i]$ gives rise to a quadratic Dirichlet
$L$-function over Gaussian integers upon abusing notation of ideals as in
[BBCL22, Section 2.1]:

$L(s,\chi_{D})\coloneqq\sum_{n\neq
0}\frac{\chi_{D}(n)}{\mathrm{N}(n)^{s}},\qquad\mathrm{Re}(s)>1.$

If $D$ is squarefree, then $\chi_{D}$ is primitive and $L(s,\chi_{D})$ extends
to a holomorphic function with a functional equation relating the values at
$s$ and $1-s$. The generalised Riemann hypothesis then asserts that all the
nontrivial zeros would lie on the critical line $\mathrm{Re}(s)=\frac{1}{2}$.
As a weaker version thereof, the generalised Lindelöf hypothesis in the hybrid
aspect asserts that $\theta=0$ would hold in (1.4). If $\theta<\frac{1}{4}$ is
valid, then (1.4) is said to be subconvex in the hybrid aspect.

Given $m,n,c\in\mathbb{Z}[i]$ with $c\neq 0$, we define Kloosterman sums over
Gaussian integers by

$\mathcal{S}(m,n;c)\coloneqq\sum_{a\in(\mathbb{Z}[i]/(c))^{\times}}e(\langle
m,a/c\rangle+\langle n,a^{\ast}/c\rangle),$

where $a^{\ast}$ denotes the multiplicative inverse of $a$ modulo $(c)$, i.e.
$aa^{\ast}\equiv 1{\@displayfalse\pmod{c}}$, and $\langle x,y\rangle$ denotes
the standard inner product on $\mathbb{R}^{2}\cong\mathbb{C}$. Then the
Kloosterman sums over Gaussian integers obey the Weil bound (see [Mot97,
Equation (3.5)])

$|\mathcal{S}(m,n;c)|\leq\sqrt{\mathrm{N}(c)}|(m,n,c)|\tau(c),$ (2.1)

where $\tau(c)$ is the number of divisors of $c$ and $\mathrm{N}(c)=|c|^{2}$
is the norm form. Furthermore, let $\rho(c,a)$ be the number of solutions
$d{\@displayfalse\pmod{c}}$ of

$d^{2}-ad+1\equiv<EMAIL_ADDRESS>
By definition, we derive an expression à la Iwaniec [Iwa84a, Equation (14)] of
Kloosterman sums with $m=n$ as a sum of additive characters weighted by the
counting function $\rho(c,a)$.

######  Lemma 2.1.

Keep the notation as above. Then we have that

$\mathcal{S}(n,n;c)=\sum_{a\in(\mathbb{Z}[i]/(c))^{\times}}\rho(c,a)e(\langle
n,a/c\rangle).$

Lemma 2.1 plays a key role in treating the arithmetic side of the Kuznetsov
formula in the methods of Iwaniec [Iwa84a] and Cai [Cai02], subsequently
enhanced by the author [Kan24].

##  3\. Auxiliary Lemmata

This section prepares for the proof of Theorem 1.1.

###  3.1. Oscillatory Integrals

Let

$\mathcal{I}\coloneqq\int_{a}^{b}w(t)e^{i\phi(t)}dt,$ (3.1)

where $w$ and $\phi$ are real-valued smooth functions on $\mathbb{R}$. A
fundamental estimate for integrals of the shape (3.1) is the $r$-th derivative
test

$\mathcal{I}\ll\underset{[a,b]}{\operatorname{\mathrm{Var}}}\,w(t)\left(\min_{[a,b]}|\phi^{(r)}(t)|^{\frac{1}{r}}\right)^{-1}.$

Nonetheless, our analysis requires sharper estimates in an asymptotic form as
in [BKY13, Section 8], which we formulate in the language of inert functions
introduced in [KPY19].

######  Definition 3.1.

Let $\mathcal{F}$ be an indexing set, and let
$X\colon\mathcal{F}\to[1,\infty)$ be a function of $T\in\mathcal{F}$ so that
$X_{T}\in[1,\infty)$. A family $\\{w_{T}\\}_{T\in\mathcal{F}}$ of smooth
functions supported on a product of dyadic intervals in $\mathbb{R}^{d}_{+}$
is called $X$-inert if for each $j=(j_{1},\ldots,j_{d})\in\mathbb{N}_{0}^{d}$,
we have that

$\sup_{T\in\mathcal{F}}\sup_{(x_{1},\ldots,x_{d})\in\mathbb{R}^{d}_{+}}X_{T}^{-(j_{1}+\cdots+j_{d})}|x_{1}^{j_{1}}\cdots
x_{d}^{j_{d}}w_{T}^{(j_{1},\ldots,j_{d})}(x_{1},\ldots,x_{d})|\ll_{j_{1},\ldots,j_{d}}1.$

######  Lemma 3.2.

Suppose that $w=w_{T}$ is a family of $X$-inert functions with compact support
on $[1,2]$ such that $w^{(j)}(t)\ll_{j}X^{j}$. Suppose that $\phi$ is smooth
and satisfies $\phi^{(j)}(t)\ll YZ^{-j}$ for some $YX^{-2}\geq R\geq 1$ and
$t\in\mathrm{supp}(w)$.

  1. (i)

If $|\phi^{\prime}(t)|\gg YZ^{-1}$ for $t\in\mathrm{supp}(w)$, then
$\mathcal{I}\ll_{A}ZR^{-A}$ for an arbitrarily large $A>0$. Furthermore, the
same statement holds under the weaker condition $YX^{-1}\geq R$.

  2. (ii)

If $\phi^{\prime\prime}(t)\gg YZ^{-2}$ for $t\in\mathrm{supp}(w)$ and there
exists some (not necessarily unique) $t_{0}\in\mathbb{R}$ such that
$\phi^{\prime}(t_{0})=0$, then we have for an arbitrarily large $A>0$ that

$\mathcal{I}=\frac{e^{i\phi(t_{0})}}{\sqrt{\phi^{\prime\prime}(t_{0})}}F_{T}(t_{0})+O_{A}(ZR^{-A}),$

where $F_{T}$ is a family of $X$-inert functions supported on $t_{0}\asymp Z$.

  3. (iii)

Keep the assumptions as in (ii). Let $U\gg
R^{\varepsilon}\phi^{\prime\prime}(t_{0})^{-\frac{1}{2}}$, and let $W_{0}\in
C_{c}^{\infty}([-2,2])$ be such that $W_{0}(x)=1$ for $|x|\leq 1$. Then we
have for an arbitrarily large $A>0$ that

$\mathcal{I}=\int_{-\infty}^{\infty}w(t)W_{0}\left(\frac{t-t_{0}}{U}\right)e^{i\phi(t)}dt+O_{A}(R^{-A}).$

###  3.2. Character Sums

Character sums over algebraic number fields in general are studied by Hinz
[Hin83a, Hin83b, Hin86] among other number theorists, which generalise
classical Burgess bounds [IK04, Theorem 12.6]; cf. [Bur62, Bur63]. The author
[Kan24, Lemma 2.1] incorporates hybrid Weyl-strength subconvex bounds for
Dirichlet $L$-functions with Burgess bounds to prove bootstrapped Burgess
bounds that are of independent interest. Nonetheless, the proof of Theorem 1.1
builds on the case where $r\to\infty$ as in loc. cit. It thus suffices for our
purposes to handle character sums in short intervals trivially with absolute
values, and then input subconvex bounds at hand. Establishing bootstrapped
Burgess bounds in short intervals over number fields for all $r\in\mathbb{N}$
would involve some brute force calculations.

In anticipation of the subsequent analysis, we record an analogue of [Kan24,
Lemma 2.3].

######  Lemma 3.3.

For a fundamental discriminant $D$, let $\chi_{D}=(\frac{D}{\cdot})$ be the
primitive quadratic character modulo $(D)$ over $\mathbb{Q}(i)$, and suppose
that the hybrid subconvex bound

$L\left(\frac{1}{2}+it,\chi_{D}\right)\ll_{\varepsilon}(\mathrm{N}(D)(1+|t|))^{\theta+\varepsilon}$
(3.2)

holds for some $0\leq\theta<\frac{1}{4}$. Then we have for any $x\geq 1$ and
$\varepsilon>0$ that

$\sum_{\mathrm{N}(n)\leq
x}\left(\frac{D}{n}\right)\ll_{\varepsilon}x^{\alpha}\mathrm{N}(D)^{\beta+\varepsilon},$
(3.3)

where

$\alpha=1-\frac{3}{2(2+\theta)},\qquad\beta=\frac{2\theta}{2+\theta}.$ (3.4)

###### Proof.

The proof strategy follows verbatim the argument of the author [Kan24, Lemma
2.1]. Let $0<h<x$ be a parameter at our disposal. Let $w$ be a smooth weight
function so that $w(t)=1$ for $0\leq t\leq\sqrt{x}$, $w(t)=0$ for
$t\geq\sqrt{x+h}$, and satisfying $w^{(j)}(t)\ll_{j}h^{-\frac{j}{2}}$ for all
$t>0$ and $j\in\mathbb{N}_{0}$. Following the notation of Petrow–Young [PY23,
Page 1889], we define

$\mathcal{S}(\chi_{D},w)\coloneqq\sum_{n\neq
0}\left(\frac{D}{n}\right)w(|n|)=\frac{1}{2\pi
i}\int_{(\sigma)}\widetilde{w}(s)L(s,\chi_{D})ds.$

Integration by parts allows for the truncation of the integral up to
$\mathrm{Im}(s)\ll(\frac{x}{h})^{\frac{1}{2}+\varepsilon}$. Choosing
$\sigma=\frac{1}{2}$ and using the assumption (3.2) then yields

$\mathcal{S}(\chi_{D},w)\ll_{\varepsilon}x^{\frac{1}{4}}\mathrm{N}(D)^{\theta+\varepsilon}\left(\frac{x}{h}\right)^{\frac{\theta}{2}+\varepsilon}.$

We observe that

$\sum_{\mathrm{N}(n)\leq
x}\left(\frac{D}{n}\right)=\mathcal{S}(\chi_{D},w)-\sum_{x<\mathrm{N}(n)\leq
x+h}\left(\frac{D}{n}\right)w(n).$

Bounding the sum on the right-hand side trivially with absolute values, we
deduce

$\sum_{\mathrm{N}(n)\leq
x}\left(\frac{D}{n}\right)\ll_{\varepsilon}x^{\frac{1}{4}}\mathrm{N}(D)^{\theta+\varepsilon}\left(\frac{x}{h}\right)^{\frac{\theta}{2}+\varepsilon}+h.$

Optimising
$h=x^{1-\frac{3}{2(2+\theta)}}\mathrm{N}(D)^{\frac{2\theta}{2+\theta}}$
concludes the proof of Lemma 3.3. ∎

###  3.3. Mean Value Estimates

Given $1\leq A\leq|B|$ and $C\geq 1$, we define

$\mathcal{F}(A,B,C)\coloneqq\sum_{B\leq\mathrm{N}(a)\leq
A+B}\sum_{C\leq\mathrm{N}(c)\leq 2C}\rho(c,a).$

The following asymptotic formula serves as an algebraic counterpart of [Kan24,
Lemma 2.2].

######  Lemma 3.4.

Let $1\leq A\leq|B|$ and $C\geq 1$. Suppose that (3.3) holds for some
$0\leq\alpha\leq 1$ and $0\leq\beta\leq\frac{1}{2}$. Then there exists an
absolute and effectively computable constant $\lambda>0$ such that

$\mathcal{F}(A,B,C)=\lambda
AC+O_{\varepsilon}((A^{\frac{1}{3-2\alpha}}|B|^{\frac{2\beta}{3-2\alpha}}C^{\frac{3}{2}-\frac{1}{3-2\alpha}}+AC^{\frac{1}{2}})(|B|C)^{\varepsilon}).$

###### Proof.

The proof proceeds as in [Kan24] mutatis mutandis. The only requisite
adjustment is a replacement of all the integers therein with the norms of
nonzero ideals of $\mathbb{Z}[i]$. We also employ Lemma 2.1 in place of the
fifth last display on [Kan24, Page 5]. Note that the leading coefficient
$\lambda$ can be explicitly determined as in two dimensions, but there is no
necessity for our purposes so we leave $\lambda$ unspecified. The proof of
Lemma 3.4 is thus complete. ∎

##  4\. Koyama’s Machinery

As a preparation for the proof of Theorem 1.1, we incorporate the automorphic
techniques of Iwaniec [Iwa84a] with the machinery of Koyama [Koy01], the
latter being a $3$-dimensional analogue of the machinery of Luo–Sarnak [LS95].
A key difference between these two lies in a treatment of Kloosterman sums.
The former features an expression of Kloosterman sums in term of a certain
counting function and reduces the analysis to the estimation of quadratic
character sums, while the latter features the Weil bound for Kloosterman sums.
Notably, our argument detects via the stationary phase analysis an oscillatory
factor hidden in an integral of Bessel functions. This makes Lemma 3.3
applicable building on the notion of Cai [Cai02].

######  Lemma 4.1.

Let $N,T,X\geq 1$ with $N\ll T^{2}$. Suppose that (3.3) holds for some
$0\leq\alpha\leq 1$ and $0\leq\beta\leq\frac{1}{2}$. Then we have for any
$\varepsilon>0$ that

$\sum_{|t_{j}|\leq
T}X^{it_{j}}\ll_{\varepsilon}N^{-\frac{1}{3-2\alpha}}T^{\frac{1}{2}+\frac{1}{3-2\alpha}}X^{\frac{1}{2}+\frac{2\beta}{3-2\alpha}+\varepsilon}+T^{3+2\theta+\varepsilon}N^{-\frac{1}{2}}.$
(4.1)

######  Remark 4.2.

The restriction $N\ll T^{2}$ ensures that the contribution of the spectral
side of the Kuznetsov formula [Mot96, Mot97, Mot01] is negligibly small
compared to the arithmetic side. This phenomenon is in harmony with the
implicit restriction on $N$ imposed in [Cai02].

###### Proof.

We begin with the argument of Koyama [Koy01]. The threshold of improvement is
on the third last display on [Koy01, Page 790], which we rewrite (with minor
adjustments to the notation) as

$\frac{1}{2\pi}\sum_{c\neq
0}\frac{\mathcal{S}(n,n;c)}{\mathrm{N}(c)}\sum_{\ell=0}^{\infty}\sum_{m=0}^{\infty}\frac{(-1)^{\ell+m}}{\ell!m!}\left(\frac{z}{2}\right)^{2\ell}\left(\frac{\overline{z}}{2}\right)^{2m}\\\
\times\int_{-\infty}^{\infty}\frac{t^{2}}{\sinh(\pi
t)}\frac{e^{\pi|t|}}{|t|^{\ell+m+1}}\left|\frac{z}{2t}\right|^{2it}X^{i|t|}e^{-\frac{|t|}{T}}(1+O(|t|^{-1}))dt,$
(4.2)

where $z=\frac{2\pi n}{c}$. In view of the observation on [Koy01, Page 792],
we eliminate in advance the sums over $\ell$ and $m$ because the contribution
of $\ell=m=0$ dominates the others. Hence, the integral in (4.2) boils down
essentially to (cf. [Koy01, Equation (4.6)])

$\int_{0}^{\infty}\left(\frac{|n|X^{\frac{1}{2}}}{|c|t}\right)^{2it}te^{-\frac{t}{T}}(1+O(t^{-1}))dt.$

Setting $Y=\frac{|n|X^{\frac{1}{2}}}{|c|}$ and making a change of variables
$t\to tY$ yields

$Y^{2}\int_{0}^{\infty}te^{-2itY\log t-\frac{tY}{T}}(1+O((tY)^{-1}))dt.$

By Lemma 3.2 (ii), this is asymptotically equal (up to an absolute constant)
to

$Y^{\frac{3}{2}}e^{\frac{2iY}{e}-\frac{Y}{eT}}(1+O(Y^{-1})),$ (4.3)

which decays exponentially unless
$\mathrm{N}(c)\gg\frac{\mathrm{N}(n)X}{(T\log T)^{2}}$. Koyama [Koy01, Page
791] bounds (4.3) with absolute values and analyse the resulting sum over $c$
via the Weil bound, but we make use of an additional oscillatory factor
involved in (4.3) for further cancellations.

This is where the idea of Iwaniec [Iwa84a] and Cai [Cai02] comes into play.
More precisely, Lemma 2.1 implies that (4.2) averaged over $n$ is
asymptotically equal (once again up to an absolute constant and a negligibly
small error term) to

$\frac{X^{\frac{3}{4}}}{N}\sum_{\mathrm{N}(c)\gg\frac{NX}{(T\log
T)^{2}}}\frac{1}{\mathrm{N}(c)^{\frac{7}{4}}}\sum_{a\in(\mathbb{Z}[i]/(c))^{\times}}\rho(c,a)\sum_{n}h(n)\mathrm{N}(n)^{\frac{3}{4}}e(\langle
n,(a+X^{\frac{1}{2}})/c\rangle).$ (4.4)

Here $h:(0,\infty)\to\mathbb{R}$ is a fixed smooth compactly supported
function with holomorphic Mellin transform
$\widetilde{h}:\mathbb{C}\to\mathbb{C}$. We choose $h$ such that it is
supported on a dyadic interval $[\sqrt{N},\sqrt{2N}]$ whose derivatives
satisfy

$h^{(j)}(\xi)\ll_{j}N^{-\frac{j}{2}},\qquad j\in\mathbb{N}_{0},$

and whose mean value is

$\int_{-\infty}^{\infty}h(\xi)\xi d\xi=\widetilde{h}(2)=N.$

The expression (4.4) is in harmony with [Cai02, Equation (6.1)] where
$2\,\mathrm{Re}(\cosh\beta)\asymp X^{\frac{1}{2}}$ in his notation. Via a
dyadic subdivision, we break up the sum over $c$ into at most $O(\log X)$
subsums over $C\leq\mathrm{N}(c)\leq 2C$ with $C\coloneqq\frac{NX}{(T\log
T)^{2}}$. Applying Poisson summation in $n$ implies

$\sum_{n}h(n)\mathrm{N}(n)^{\frac{3}{4}}e(\langle
n,(a+X^{\frac{1}{2}})/c\rangle)=\int_{-\infty}^{\infty}h(x)x^{\frac{3}{2}}e(\langle
x,(a+X^{\frac{1}{2}})/c\rangle)dx\\\ +\sum_{n\neq
0}\int_{-\infty}^{\infty}h(x)x^{\frac{3}{2}}e(\langle
x,(a+X^{\frac{1}{2}})/c\rangle)e(-nx)dx.$

Integration by parts shows that the contribution of the second term is
bounded. Furthermore, repeated integration by parts once again shows that the
first term is also bounded whenever $|a+X^{\frac{1}{2}}|\geq
C^{\frac{1}{2}}N^{-\frac{1}{2}+\varepsilon}\eqqcolon A$. Hence, we arrive at
(after making a change of variables $x\to|c|x$)

$\frac{X^{\frac{3}{4}}}{C^{\frac{1}{2}}N}\int_{-\infty}^{\infty}\sum_{C\leq\mathrm{N}(c)\leq
2C}c\sum_{|a+B|\leq A}\rho(c,a)e(\langle
x,a+B\rangle)h(|c|x)x^{\frac{3}{2}}dx,$

where we define $B\coloneqq X^{\frac{1}{2}}$ for notational simplicity. In
order to apply Lemma 3.4, we invoke $\mathrm{N}(a)=|a|^{2}$, and take $A\to
C^{\frac{1}{2}}N^{-\frac{1}{2}}X^{\frac{1}{2}+\varepsilon}$ and $B\to X$ in
Lemma 3.4. Therefore, if we define

$\displaystyle\mathcal{F}_{x}(A,B,C)$
$\displaystyle\coloneqq\sum_{C\leq\mathrm{N}(c)\leq 2C}\sum_{|a+B|\leq
A}\rho(c,a)e(\langle x,a+B\rangle)$ $\displaystyle=\frac{\lambda C\sin(2\pi
Ax)}{\pi
x}+O_{\varepsilon}((1+Ax)(N^{-\frac{1}{2(3-2\alpha)}}X^{\frac{1+4\beta}{2(3-2\alpha)}}C^{\frac{3}{2}-\frac{1}{2(3-2\alpha)}}+CN^{-\frac{1}{2}}X^{\frac{1}{2}})X^{\varepsilon}),$

then (cf. [Cai02, Page 70])

$\displaystyle\sum_{C\leq\mathrm{N}(c)\leq 2C}\sum_{|a+B|\leq
A}\rho(c,a)e(\langle x,a+B\rangle)h(|c|x)$
$\displaystyle\qquad=\int_{C}^{2C}h(xy)d_{y}\mathcal{F}_{x}(A,B,y)$
$\displaystyle\qquad=\frac{\lambda\sin(2\pi Ax)}{\pi
x}\int_{C}^{2C}h(xy)dy+O_{\varepsilon}((N^{-\frac{1}{2(3-2\alpha)}}X^{\frac{1+4\beta}{2(3-2\alpha)}}C^{\frac{3}{2}-\frac{1}{2(3-2\alpha)}}+CN^{-\frac{1}{2}}X^{\frac{1}{2}})X^{\varepsilon}).$

The contribution of the first term is negligibly small by making a change of
variables coupled with repeated integration by parts. Because $h$ is supported
on $[\sqrt{N},\sqrt{2N}]$ and $C=\frac{NX}{(T\log T)^{2}}$, integrating over
$x$ trivially implies that the contribution of the second term amounts to

$\frac{X^{\frac{3}{4}}}{C^{\frac{1}{2}}N}\cdot\frac{N^{\frac{5}{4}}}{C^{\frac{5}{4}}}(N^{-\frac{1}{2(3-2\alpha)}}X^{\frac{1+4\beta}{2(3-2\alpha)}}C^{\frac{3}{2}-\frac{1}{2(3-2\alpha)}}+CN^{-\frac{1}{2}}X^{\frac{1}{2}})X^{\varepsilon}\ll_{\varepsilon}N^{-\frac{1}{3-2\alpha}}T^{\frac{1}{2}+\frac{1}{3-2\alpha}}X^{\frac{1}{2}+\frac{2\beta}{3-2\alpha}+\varepsilon}.$

This concludes our analysis of sums of Gaussian Kloosterman sums on the
arithmetic side of the Kuznetsov formula, and the contribution of the spectral
side is negligible; see Remark 4.2.

The second term in (4.1) stems from a mean value estimate for Rankin–Selberg
$L$-functions on the critical line. In this respect, the result of
Balkanova–Frolenkov [BF22, Theorem 1.3] shows333A transition from the second
moment of symmetric square $L$-functions to the first moment of Rankin–Selberg
$L$-functions is fundamental; see for instance the second last display on
[Kan22, Page 1851]. the exponent $3+2\vartheta+\varepsilon$, where $\vartheta$
is defined in (1.2). Note that Koyama [Koy01] assumes the mean-Lindelöf
hypothesis, and hence a fortiori the generalised Lindelöf hypothesis
$\vartheta=0$. Because $\vartheta$ is replaceable with $\theta$ by definition,
the proof of Lemma 4.1 is complete. ∎

It is convenient to restate Lemma 4.1 upon substituting the values of $\alpha$
and $\beta$ in (3.4).

######  Lemma 4.3.

Let $N,T,X\geq 1$ with $N\ll T^{2}$. Then we have for any $\varepsilon>0$ that

$\sum_{|t_{j}|\leq
T}X^{it_{j}}\ll_{\varepsilon}N^{-1+\frac{3}{5+\theta}}T^{\frac{3(3+\theta)}{2(5+\theta)}}X^{\frac{9}{2}-\frac{20}{5+\theta}+\varepsilon}+T^{3+2\theta+\varepsilon}N^{-\frac{1}{2}}.$

##  5\. Proof of Theorem 1.1

We now embark on the proof of Theorem 1.1. In Lemma 4.3, we optimise
$N=T^{2}$, which is akin to the choice in the last display on [Cai02, Page
67], obtaining

$\sum_{|t_{j}|\leq
T}X^{it_{j}}\ll_{\varepsilon}T^{-\frac{1}{2}+\frac{3}{5+\theta}}X^{\frac{9}{2}-\frac{20}{5+\theta}+\varepsilon}+T^{2+2\theta+\varepsilon}.$
(5.1)

Nonetheless, unlike in two dimensions, the size of our spectral exponential
sums and the error term in the smooth explicit formula [BBCL22, Lemma 4.1] is
not so well-balanced that the latter may dominate the former. To counterweigh
these contributions, we worsen the quality of the first term in (5.1) on
purpose, and introduce $\delta>0$ as an additional exponent of $X$.

Henceforth, we restrict our attention to

$X^{\varepsilon}\ll T\ll
X^{\frac{5+9\theta+2\delta(5+\theta)}{19+\theta(25+4\theta)}}.$

Then the worsened first term on the right-hand side of (5.1) absorbs the
second term, thus

$\sum_{|t_{j}|\leq
T}X^{it_{j}}\ll_{\varepsilon}T^{-\frac{1}{2}+\frac{3}{5+\theta}}X^{\frac{9}{2}-\frac{20}{5+\theta}+\delta+\varepsilon}.$
(5.2)

We invoke the second display on [BCC+19, Page 5363] with $\alpha=2\theta$ by
[BF22, Theorem 1.3]:

$\sum_{|t_{j}|\leq
T}X^{it_{j}}\ll_{\varepsilon}T^{\frac{7+4\theta}{4}}X^{\frac{1}{4}+\varepsilon}+T^{2}.$
(5.3)

This enables an interpolation between (5.2) and (5.3) such that the exponent
of $T$ is adjusted to $1$. Applying the inequality $\min(A,B)\leq
A^{\gamma}B^{1-\gamma}$ for $A,B>0$ and $0<\gamma<1$ yields

$\sum_{|t_{j}|\leq
T}X^{it_{j}}\ll_{\varepsilon}\min(T^{-\frac{1}{2}+\frac{3}{5+\theta}}X^{\frac{9}{2}-\frac{20}{5+\theta}+\delta+\varepsilon},T^{\frac{7+4\theta}{4}}X^{\frac{1}{4}+\varepsilon})\ll_{\varepsilon}TX^{\frac{12+\delta(5+\theta)(3+4\theta)+\theta(25+18\theta)}{33+\theta(29+4\theta)}+\varepsilon}$
(5.4)

for $\gamma=1-\frac{6(3+\theta)}{33+\theta(29+4\theta)}$. It is
straightforward to verify that the contribution of the second term $T^{2}$ in
(5.3) is smaller than (5.4). The bound (5.4) is thought of as a
$3$-dimensional counterpart of [Kan24, Equation (3.4)]. Furthermore, it is
crucial to our analysis that the choice of $\gamma$ is independent of
$\delta>0$ because the way we interpolate pivots upon the exponents of $T$. It
will become clear in the ensuing discussion why we optimise $\gamma$ in such a
way that the resulting exponent of $T$ is adjusted to $1$.

We now borrow the smooth explicit formula [BBCL22, Lemma 4.1] due to the
unbreakable barrier $O_{\varepsilon}(X^{\frac{3}{2}+\varepsilon})$ in the
explicit formula of Nakasuji [Nak00, Nak01]. Let $Y\in[X^{\frac{1}{2}},X]$ be
a parameter to be chosen later, and let $k:(0,\infty)\to[0,\infty)$ be a
smooth real-valued function with compact support on $[Y,2Y]$. Suppose that
$\int_{-\infty}^{\infty}k(u)du=1$ and that for all $j\in\mathbb{N}_{0}$,

$\int_{-\infty}^{\infty}|k^{(j)}(u)|du\ll_{j}Y^{-j}.$

By the Bykovskiĭ-type theorem [BBCL22, Theorem 1.3] and routine computations
in [Kan22, Equation (3-64)], we deduce in a shorter regime
$Y\in[X^{\frac{1}{3}+\varepsilon},X]$ that

$\mathcal{E}_{\Gamma}(X)=E(X;k)+O_{\varepsilon}(X^{\frac{4\theta+6}{5}+\varepsilon}Y^{\frac{2}{5}}+X^{1+\varepsilon}),$
(5.5)

where

$E(X;k)\coloneqq\sum_{|t_{j}|\leq\frac{X^{1+\varepsilon}}{Y}}\frac{1}{s_{j}}\int_{Y}^{2Y}(X+u)^{s_{j}}k(u)du,\qquad
s_{j}=1+it_{j}.$

Note that the first term in the error term in (5.5) absorbs the second term
regardless of how one optimises the parameter $Y$. It follows from (5.4) and
partial summation that

$E(X;k)\ll_{\varepsilon}X^{1+\frac{12+\delta(5+\theta)(3+4\theta)+\theta(25+18\theta)}{33+\theta(29+4\theta)}+\varepsilon}$
(5.6)

provided

$\frac{X^{1+\varepsilon}}{Y}\ll
X^{\frac{5+9\theta+2\delta(5+\theta)}{19+\theta(25+4\theta)}}.$

We need to choose $Y$ as small as possible, namely

$Y=X^{1-\frac{5+9\theta+2\delta(5+\theta)}{19+\theta(25+4\theta)}+\varepsilon}.$

Therefore, combining (5.5) and (5.6) implies

$\mathcal{E}_{\Gamma}(X)\ll_{\varepsilon}X^{1+\frac{12+\delta(5+\theta)(3+4\theta)+\theta(25+18\theta)}{33+\theta(29+4\theta)}+\varepsilon}+X^{\frac{4\theta+6}{5}+\frac{2}{5}\left(1-\frac{5+9\theta+2\delta(5+\theta)}{19+\theta(25+4\theta)}\right)+\varepsilon}.$

Consequently, the optimal value of $\delta>0$ boils down to

$\delta=\frac{411+1877\theta+2666\theta^{2}+1558\theta^{3}+552\theta^{4}+64\theta^{5}}{2085+4772\theta+3751\theta^{2}+976\theta^{3}+80\theta^{4}}.$

Given the monotonic function $\delta(\theta)$ defined by (1.5), we obtain

$\mathcal{E}_{\Gamma}(X)\ll_{\varepsilon}X^{\delta(\theta)+\varepsilon},$

as desired. The proof of Theorem 1.1 is now complete.

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