# On Edwards’ Speculation and a New Variational Method for the Zeros of the
$Z$-Function

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

In his foundational book, Edwards introduced a unique ”speculation” regarding
the possible theoretical origins of the Riemann Hypothesis, based on the
properties of the Riemann-Siegel formula. Essentially Edwards asks whether one
can find a method to transition from zeros of $Z_{0}(t)=cos(\theta(t))$, where
$\theta(t)$ is Riemann-Siegel theta function, to zeros of $Z(t)$, the Hardy
$Z$-function. However, when applied directly to the classical Riemann-Siegel
formula, it faces significant obstacles in forming a robust plausibility
argument for the Riemann Hypothesis.

In a recent work, we introduced an alternative to the Riemann-Siegel formula
that utilizes series acceleration techniques. In this paper, we explore
Edwards’ speculation through the lens of our accelerated approach, which
avoids many of the challenges encountered in the classical case. Our approach
leads to the description of a novel variational framework for relating zeros
of $Z_{0}(t)$ to zeros of $Z(t)$ through paths in a high-dimensional parameter
space $\mathcal{Z}_{N}$, recasting the RH as a modern non-linear optimization
problem.

##  1\. Introduction

The Riemann zeta function is given by $\zeta(s)=\sum_{n=1}^{\infty}n^{-s}$ in
the range $Re(s)>1$. In 1859, Bernhard Riemann published a work of monumental
importance [25], whose starting point is the analytic extension of $\zeta(s)$
to a meromorphic function with a single pole at $s=1$. This allowed him to
describe various intricate relationships between the distribution of prime
numbers and the analytical properties of $\zeta(s)$. Within this manuscript,
Riemann posited what would become one of the most famous and enduring
conjectures in mathematics:

######  Conjecture (RH).

All non-trivial zeros of $\zeta(s)$ lie on the critical line $Re(s)=1/2$.

The RH, though only briefly mentioned in Riemann’s manuscript, holds profound
implications and its resolution would represent a significant breakthrough. It
would not only provide a deeper understanding of the distribution of prime
numbers but also substantiate crucial links across various domains of
mathematics and physics with implications extending to number theory, algebra,
cryptography, and quantum physics, see [1, 27] and references therein.

However, it is noteworthy that within his ground breaking manuscript, Riemann
did not describe any methods for the practical computation of $\zeta(s)$ for
values in the complex domain, a fact that further contributed to the mystery
of the conjecture. Indeed, one comes to the zen-like question: how can one
locate the zeros of a function whose values one does not know how to
efficiently compute altogether?

For such reasons, in the early 20-th century, mathematicians such as G. H.
Hardy [9] and E. Landau [18] expressed the idea that the Riemann hypothesis
was most likely the result of a heuristic guess rather than based on a
concrete verification, not to mention on an insight into any conceptual
theoretical reason as to why one should expect it to hold.

In the 1930s, C. L. Siegel [28] uncovered previously unpublished notes by
Riemann, astonishingly showing that Riemann had actually developed
sophisticated saddle point methods for the efficient numerical evaluation of
the values of $\zeta(s)$ on the critical strip. Based on a numerical
investigation of the properties of the Riemann-Siegel formula for the first
zero of $\zeta(s)$, Edwards proposed an intriguing ”speculation” in his
seminal book [4] regarding the possible theoretical origins of the Riemann
Hypothesis. Recall that the Hardy $Z$-function is the real function defined by

$Z(t)=e^{i\theta(t)}\zeta\left(\frac{1}{2}+it\right)$ (1)

where

$\theta(t)=\text{arg}\left(\Gamma\left(\frac{1}{4}+\frac{it}{2}\right)\right)-\frac{t}{2}\log(t).$
(2)

is the Riemann-Siegel $\theta$-function [4, 13]. The Riemann Hypothesis is
equivalent to the statement that all non-trivial zeros of the Hardy
$Z$-function are real. We will refer to

$Z_{0}(t):=cos(\theta(t)),$ (3)

as _the core of $Z(t)$_. The zeros $t_{n}\in\mathbb{R}$ of the core $Z_{0}(t)$
are all real and are given as the unique solutions of

$\theta(t_{n})=\left(n+\frac{1}{2}\right)\pi,$ (4)

for any $n\in\mathbb{Z}$. Edwards’ speculation is essentially the question of
whether one can find a method to transition from zeros of $Z_{0}(t)$, which is
the first term of the Riemann-Siegel formula, to zeros of $Z(t)$.

However, according to Edwards himself, while such a method might give insight
to Riemanns’ original motivation to conjecture the RH, when applied directly
to the classical Riemann-Siegel formula, it faces significant obstacles in
forming a robust plausibility argument for the Riemann Hypothesis. Rather, it
further reaffirms its notorious difficulty rather than shed light on why it
might hold true.

In recent works [14, 15], we introduced an alternative to the Riemann-Siegel
formula via high-order sections based on series acceleration techniques. Our
aim in this work is to investigate the ideas of Edwards’ speculation applied
to our accelerated formulas, rather than to the Riemann-Siegel formula. We
show that our formula avoids many of the analytical challenges encountered in
the classical Riemann-Siegel case, leading us to introduce a new variational
framework for the study of the zeros of the zeta function.

For any $N\in\mathbb{N}$, we introduce the space $\mathcal{Z}_{N}$, of
generalized sections of Hardy’s $Z$-function

$Z_{N}(t;\overline{a})=cos(\theta(t))+\sum_{k=1}^{N}\frac{a_{k}}{\sqrt{k+1}}cos(\theta(t)-ln(k+1)t)$
(5)

where $\overline{a}=(a_{1},...,a_{N})\in\mathbb{R}^{N}$. Note that given a
path $\gamma(r)$ in the space $\mathcal{Z}_{N}$ starting at
$\gamma(0)=Z_{0}(t)$, one can continuously extend $t_{n}(r)$ to be a
corresponding zero of $Z_{N}(t;\gamma(r))$. Our main result in this work is
the following:

######  Theorem 1 (Edwards’ Speculation for High-Order Sections).

The Riemann Hypothesis holds if and only if, for any $n\in\mathbb{Z}$, there
exists a path $\gamma(r)$ in $\mathcal{Z}_{\left[\frac{t_{n}}{2}\right]}$ from
$\overline{a}=\overline{0}$ to $\overline{a}=\overline{1}$, along which
$t_{n}$ does not collide with its adjacent zeros $t_{n\pm 1}$. That is, for
which $t_{n}(r)\neq t_{n\pm 1}(r)$ for all $r\in[0,1]$.

The proof of this result is partially based on previous results [14, 15] on
Spira’s conjecture and high-order sections. In the subsequent sections, we
further expand on the crucial differences between our Theorem 1 and the
Riemann-Siegel formula setting. Mainly, we explain how, contrary to the
classical case, Theorem 1 can indeed be considered as leading to the
formulation of a meaningful plausibility argument for the RH.

The rest of this paper is organized as follows. In Section 2, we recall the
Riemann-Siegel formula and its role in numerical calculations of $Z(t)$.
Section 3 introduces the formulation of Edwards’ speculation, the core
function $Z_{0}(t)=\cos(\theta(t))$ and its zeros. It outlines three
foundational questions prompted by the speculation, each leading to in-depth
discussions in the following Sections 4 to 6

In Section 4, we outline the experimental relation between the zeros of $Z(t)$
and those of the core $Z_{0}(t)$, central to Edwards’ speculation. The
shortcomings of using the Riemann-Siegel formula for the theoretical
exploration of RH, particularly tracking the impact of terms in the Riemann-
Siegel equation on the zeros locations, are discussed in Section 5. Section 6
examines the limitations of classical numerical methods, such as the Newton
method, for effectively transitioning from zeros of $Z_{0}(t)$ to those of
$Z(t)$. Examples where the method fails to provide accurate correspondence
between the zeros are presented, highlighting the necessity for more
sophisticated approaches.

In Section 7, we present high-order sections as an alternative which avoids
the shortcomings of the classical Riemann-Siegel formula outlined in Sections
4 to 6. Building upon the high-order sections discussed in Section 7, Section
8 develops the new variational method for Edwards’ speculation, allowing to
transition from zeros of $Z_{0}(t)$ to those of $Z(t)$ in a continuous manner
via a high-dimensional parameter space and proving Theorem 1. An example of
the $730120$-th zero, for which our method is shown to successfully avoid the
limitations encountered for Newton’s method, is presented in detail in Section
9. Finally, Section 10 provides a summary of our results, concluding remarks,
and potential directions for future research.

##  2\. The Riemann-Siegel Formula - Numerics vs. Theory

In the series of works [10, 11, 12] Hardy and Littlewood developed the formula
known as the approximate functional equation. In its most widely used form the
formula gives

$Z(t)=2\sum_{k=1}^{\widetilde{N}(t)}\frac{cos(\theta(t)-ln(k)t)}{\sqrt{k}}+R(t),$
(6)

where $\widetilde{N}(t)=\left[\sqrt{\frac{t}{2\pi}}\right]$ and the error term
is given by

$R(t)=O\left(\frac{1}{\sqrt{t}}\right).$ (7)

This representation, while powerful, for many purposes, requires further
refinement of the error term $R(t)$ to enhance the level of precision, a
subject not addressed in the original works of Hardy and Littlewood.

In the 1930s, C. L. Siegel discovered the remarkable Riemann-Siegel formula by
examining Riemann’s unpublished manuscripts [28]. While the manuscripts have
been studied by various mathematicians, it was Siegel’s unique approach to
scholarship that allowed him to uncover this crucial aspect of Riemann’s work,
which had been previously unnoticed. Siegel observed that Riemann’s notes
contained a variant of the Hardy-Littlewood formula (6) but, more
astonishingly, also introduced sophisticated saddle-point techniques for the
required efficient computation and further asymptotic expansion of the error
term $R(t)$. First, the formula expresses the superb approximation

$R(t)\sim\frac{e^{-i\theta(t)}e^{-\frac{\pi
t}{2}}}{(2\pi)^{\frac{1}{2}+it}e^{-\frac{i\pi}{4}}(1-ie^{-\pi
t})}\int_{L}\frac{e^{-\widetilde{N}z}(-z)^{-\frac{1}{2}+it}}{e^{z}-1}dz,$ (8)

where the integration path $L$ encompasses the first $2\widetilde{N}(t)$
singularities of the integrand. The path begins at $-\infty$, approaches the
saddle point $z=i(2\pi t)^{1/2}$ at an angle of $\pi/4$ at the path of
steepest descent, where the integral is ”concentrated”, and then extending
back towards $\infty$. Further analysis leads to the following asymptotic
expansion to any $k$-th order

$R(t)\approx(-1)^{\widetilde{N}(t)-1}\left(\frac{t}{2\pi}\right)^{-1/4}\left(\frac{e^{i\left(\frac{t}{2}ln\left(\frac{t}{2\pi}\right)-\frac{t}{2}-\frac{\pi}{8}-\theta(t)\right)}}{1-ie^{-\pi
t}}\right)\sum_{n=0}^{k}b_{n}c_{n},$ (9)

where the $b_{n}$ coefficients are given by the following recursion relation

$b_{n+1}=\sqrt{\frac{t}{2\pi}}\frac{\pi i(2n+1)b_{n}-b_{n-2}}{4\pi^{2}(n+1)},$
(10)

with $b_{0}=1$ and $b_{-1}=0$ and the $c_{n}$ coefficients are given by

$c_{n}=\frac{n!}{2^{n}}\sum_{j=0}^{\left\lfloor n/2\right\rfloor}\frac{(2\pi
i)^{j}\Psi^{(n-2j)}(p)}{2j!(n-2j)!},$ (11)

where $p=\sqrt{\frac{t}{2\pi}}-\widetilde{N}(t)$ and
$\Psi(p):=\frac{cos(2\pi(p^{2}-p-\frac{1}{16}))}{cos(2\pi p)}$. For instance,
expanded to first-order, the Riemann-Siegel formula gives

$R(t)=(-1)^{N(t)-1}\left(\frac{t}{2\pi}\right)^{-\frac{1}{4}}\Psi(p)+O\left(\frac{1}{t^{\frac{3}{4}}}\right),$
(12)

see [4, 24]. These innovative techniques allowed Riemann to calculate the
first few non-trivial zeros of the zeta function, validating that they lie on
the critical line. In fact, since its introduction the Riemann-Siegel formula,
especially when enhanced with the Odlyzko-Schönhage algorithm, has been the
main method for numerical verification of the RH, see [8, 19, 20, 21, 22, 26,
32, 33]. The most comprehensive verification to date, achieved by Platt and
Trudigian, confirms the RH up to $3\cdot 10^{12}$, see [23]. However, up to
the present day, the Riemann-Siegel formula has primarily been regarded as a
numerical tool rather than offering a theoretical explanation for the origins
of the RH, for reasons which we will expand on in the following sections.

In particular, even after the introduction of the Riemann-Siegel formula, many
mathematicians maintained that it does not pave a reasonable direction towards
proving the RH, nor does it provide a plausible argument for its validity,
which is currently entirely considered missing. For instance, Edwards, after
describing the Riemann-Siegel formula and its proof in detail in his book [4]
follows up with a subsection entitled _”Speculations on the Genesis of the
Riemann Hypothesis”_ (Subsection 7.8), in which he writes:

> _”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)

Similarly, Atle Selberg, which was of course fluent in the theory of
computations of zeta, succinctly remarked on the absence of promising
approaches for tackling the RH, including attempts to theoretically formalize
the content of the Riemann-Siegel formula:

> _”There have probably been very few attempts at proving the Riemann
> hypothesis, because, simply, no one has ever had any really good idea for
> how to go about it!”_ (Atle Selberg)

Nevertheless, in the continuation of the same Section 7.8, as the name
suggests, Edwards presents a unique ”speculation” on the origin of the Riemann
hypothesis. In the next section we describe and analyse Edwards’ speculation
and discuss its substantial shortcomings when applied to the classical
Riemann-Siegel formula.

##  3\. The Original Formulation of Edward’s Speculation on the Riemann-Siegel
Formula and Related Questions

In Section 7.6 of his books, Edwards conducts a few sample computations via
the Riemann-Siegel formula, including the computation of the first zero of
$Z(t)$ as originally conducted by Riemann himself. Edward’s speculation, which
follows, is essentially the suggestion that the Riemann Hypothesis might have
actually originated from hands-on computational observations regarding the
inner-workings of the Riemann-Siegel formula. It is encapsulated in the
following paragraph:

_’My guess is simply that Riemann used the method followed in Section 7.6 to
locate roots and that he observed that normally — as long as $t$ is not so
large that the Riemann-Siegel formula contains too many terms and as long as
the terms do not exhibit too much reinforcement — this method allows one to go
from a zero of the first term $2\cos(\theta(t))$ to a nearby zero of $Z(t)$’_
(H. M. Edwards, 1974)

The computation of the first zero of $Z(t)$ indeed shows that it is in
proximity to the first zero of $cos(\theta(t))$. However, one may find that,
as presented, the statement remains somewhat vague, almost cryptic, and leaves
the reader with various questions. For instance, we have the following three:

Question 1 - The good:

Why should one expect the existence of a general, fundamental relationship
between the zeros of $2\cos(\theta(t))$ and those of $Z(t)$, based merely on
one numerical computation of the first zero?

Question 2 - The bad:

How exactly does one transition from zeros of $2\cos(\theta(t))$ to nearby
zeros of $Z(t)$, and what mathematically well-defined method should underpin
such a procedure?

Question 3 - the ugly:

What is the precise mathematical meaning of ”reinforcement” and of having ”too
many terms” in the Riemann-Siegel formula?

In the next sections we aim to analyse these questions and show that the
vagueness with respect to them is not accidental, each for its own reasons. In
particular, we will describe why each of the questions actually has to do with
fundamental properties of the $Z$-function and the Riemann-Siegel function.
Let us define:

######  Definition 3.1 (The core of the $Z$-function).

We refer to $Z_{0}(t)=cos(\theta(t))$ as _the core function of $Z(t)$_.

The zeros of the core $Z_{0}(t)$ are completely well-understood and their
description follows immediately from the definition:

######  Proposition 3.2 (Zeros of the core $Z_{0}(t)$).

The zeros $t_{n}\in\mathbb{R}$ of the core $Z_{0}(t)$ are all real and are
given as the unique solutions of

$\theta(t_{n})=\left(n+\frac{1}{2}\right)\pi,$ (13)

for any $n\in\mathbb{Z}$.

We start with the good part, the expectation for the existence of a general
relation between the zeros of the core $Z_{0}(t)$ and those of $Z(t)$ itself.

##  4\. The Relation of the Zeros of $Z_{0}(t)$ and the Zeros of $Z(t)$

Since the idea to transition from a zero of $Z_{0}(t)$ to a zero of $Z(t)$ is
based on the computation of the first zero of $Z(t)$ via the Riemann-Siegel
formula, it would be properly described as an experimental observation. As an
experimental observation it would greatly benefit from further examples,
beyond the one presented by Edwards, for a general pattern to be observed. The
following Fig. 1 shows the graphs111$\ln|F(t)|$ is taken to accentuate the
zeros occurring where the function converges to $-\infty$. of $ln\lvert
Z(t)\rvert$ (blue) compared to $ln\lvert Z_{0}(t)\rvert$ (orange) for $0\leq
t\leq 50$:

Figure 1. Graph of $ln\lvert Z(t)\rvert$ (blue) and $ln\lvert Z_{0}(t)\rvert$
(orange) for $0\leq t\leq 50$.

Figure 1 indeed reveals the remarkable phenomenon Edwards is hinting at,
showcasing what might be referred to as one of the most fundamental ”facts of
nature” regarding the zeros of $Z(t)$. The observed phenomenon is that the
zeros of the core $Z_{0}(t)$, which is the leading term of (6), seem to
provide an initial, rough, approximation of the zeros of $Z(t)$.

However, although further vast numerical verification shows that this
phenomenon extends well beyond the range presented in Fig. 1, it remains,
fundamentally, strictly an _experimental_ phenomenon rooted in empirical
evidence rather than a formal mathematical statement. In fact, the crux of
Edwards’ speculation can be described as the question of whether the observed
empirical relationship between the zeros of $Z_{0}(t)$ and $Z(t)$ can be
formalized in a mathematically valid manner, a question of monumental
importance for the study of the Riemann Hypothesis.

######  Remark 4.1 (Franca-LeClair).

It is reasonable to suggest that the relationship between the zeros of
$Z_{0}(t)$ and $Z(t)$, as an empirical phenomenon, was discovered or even
folklorically known, at times, to various mathematicians along the years, and,
adopting Edwards’ perspective, it was already probably observed by Riemann
himself. For instance, Spira [30] studied a variant of the core, given by
$\zeta_{0}(s):=1+\chi(s)$, which in his case arises as the first section of
the Hardy-Littlewood formula for $\zeta(s)$, and proved that all of its zeros
lie on the critical line. Essentially, the zeros of $\zeta_{0}(s)$ are given
by $s_{n}=\frac{1}{2}+it_{n}$ where $t_{n}$ are the zeros of $Z_{0}(t)$
described in Proposition 3.2.

More recently, the function $1+\chi(s)$ arose in the investigations of Franca-
LeClair [5, 6]. Interestingly, in their case, they came to study $1+\chi(s)$
in a way that is completely independent of the Riemann-Siegel formula, but
through their theory of transcendental equations for zeros of zeta. Franca and
LeClair also conducted vast further numerical investigations on the relation
between zeros of $\zeta_{0}(s)$ and $\zeta(s)$, uncovering various intriguing
additional relations. They also observe that the values of $t_{n}$ can be
expressed in closed form as follows:

$t_{n}=\frac{(8n-11)\pi}{4W_{0}\left(\frac{8n-11}{8e}\right)},$ (14)

where $W_{0}(x)$ represents the principal branch of the Lambert W function,
for any $n\in\mathbb{Z}$. It is noteworthy that, although it follows directly
from the definition of $W_{0}(x)$, such a formulation could not have been
presented in earlier works. This is because the Lambert $W$-function, despite
being historically studied, was only recently formalized as a special function
[2].

Interestingly, Franca and LeClair come to a variant of Edwards’ question,
seeking a method to relate $t_{n}$ to the zeros of zeta. However, as we will
see, their approach encounters challenges similar to those in Edwards’ method
using the Riemann-Siegel formula, particularly in the ability to describe such
a method in practice, beyond an empirical observation.

In order to proceed, the question is, as Edwards points out - how do we
transition from $Z_{0}(t)$ to $Z(t)$ in a mathematically valid manner?
Naturally, such a transition requires a formula for computing $Z(t)$, with the
Riemann-Siegel formula being the immediate candidate. This introduces the
second central issue in Edwards’ speculation, which critically prevents it
from leading to a plausible argument for the RH.

##  5\. The Infeasibility of Tracking the Effect of the Terms of the Riemann-
Siegel Equation

If one wants to proceed to study how zeros of $Z_{0}(t)$ transition to zeros
of $Z(t)$ via the Riemann-Siegel formula, the method is sort of self-evident
in theory. We ”just” need to track down the contribution of the two components
comprising the formula:

1) The Hardy-Littlewood AFE:

Since $Z_{0}(t)=\cos(\theta(t))$ is the first term of the Hardy-Littlewood AFE
(6), we need to track how the addition of the rest of the terms of the
approximate functional equation impacts the location of the zeros. These
additional terms are of the form $\frac{\cos(\theta(t)-\ln(k)t)}{\sqrt{k}}$
for $k=2,\ldots,\widetilde{N}(t)$, where the number of terms,
$\widetilde{N}(t)-1$, increases to infinity as $t$ grows.

2) The Riemann-Siegel error:

Assuming the impact of the terms from (6) on a given zero $t_{n}$ of the core
$Z_{0}(t)$ can be tracked, we would then need to turn to an analysis of the
effect of the Riemann-Siegel error (9). From a numerical point-of-view, this
stage is actually less costly than the former, and the coefficients $b_{n}$
and $c_{n}$ can be efficiently computed using standard symbolic computation
software.

However, from an analytic point-of-view, the ”ugly truth” is that a tremendous
challenge arises from the fact that $b_{n}$ and $c_{n}$ lack closed-form
expressions. Specifically, $b_{n}$ is defined via a recursive formula, and
$c_{n}$ is reliant on high-order derivatives of $\Psi(p)$, rendering the task
theoretically completely infeasible.

Hence, from a modern perspective, the main challenge in studying the Riemann
Hypothesis arguably stems from the fact that, while the Riemann-Siegel formula
enables efficient numerical computations of $Z(t)$ and its zeros, it actually
provides minimal analytical insight into their structure. This issue is
primarily, but not solely, attributed to the absence of closed-form
expressions for the coefficients $c_{n}$ and $b_{n}$ involved in estimating
the error $R(t)$.

To further illustrate the magnitude of the challenge, it’s important to note
that for many zeros $t_{n}$, the main term of the AFE (6) might be
sufficiently accurate for observing the RH. However, for certain zeros
$t_{n}$, the AFE alone falls short of providing adequate verification for the
hypothesis. For such zeros the evaluation of the error term (9) becomes
crucial. The following Fig. 2 displays $\ln\left|Z(t)\right|$ (orange)
alongside the AFE approximation (blue) within the range $412<t<419$:

Figure 2. Graphs of $ln\lvert Z(t)\rvert$ (orange) and the AFE approximation
(blue) in the range $412<t<419$

The figure illustrates an instance of two real zeros of $Z(t)$ missed by the
Hardy-Littlewood formula (highlighted in red). Actually the AFE formula
erroneously identifies these two zeros as complex zeros, and hence is not
sensitive enough in order to verify the RH by itself. In fact, It was already
observed by Siegel himself [28] that for any order $k\in\mathbb{N}$ there will
be an infinite number of zeros of $Z(t)$ which will require the evaluation of
the error term $R(t)$ to an order of at least $k$. In particular, even the
question of determining to what order of precision $k=k(n)$ the Riemann-Siegel
error term (9) must be evaluated to confirm the validity of the RH for the
$n$-th zero of $Z(t)$ is completely open.

Hence, while the Riemann-Siegel formula is the main tool for evaluating $Z(t)$
numerically, the characteristics outlined in this section describe what seems
to be unbreachable obstacles to any theoretical advancement on the RH through
the formula. Specifically, a more manageable alternative to the Riemann-Siegel
formula is crucial in order for new analytical progress to be made on the
study of zeros of $Z(t)$.

##  6\. The Failure of the Classical Newton Method to Transition from Zeros of
$Z_{0}(t)$ to Zeros of $Z(t)$

Stating the obvious, the RH concerns the zeros of the equation $Z(t)=0$. In
practice, finding closed-form solutions for equations of the form $F(t)=0$ is
rare, and given the transcendental nature of $Z(t)$, there is no expectation
of a closed-form expression for its zeros similar to the one given in (14) for
$Z_{0}(t)$.

For example, in the case of algebraic functions where $F(t)$ is a general
polynomial, the Abel-Ruffini theorem states that roots can be explicitly
solved in closed form only if the degree of $F(t)$ is less than five. However,
for higher degrees, the zeros can only be located via numerical methods, such
as the Newton method. As a starting point, the method requires a rough initial
guess $t=t^{1}$ for a zero of a function $F(t)$, then applies the iterative
process

$t^{k+1}=t^{k}-\frac{F(t^{k})}{F^{\prime}(t^{k})},$ (15)

aiming to locate a true zero, given by
$\widetilde{t}=\lim_{k\rightarrow\infty}(t^{k})$, see for instance [3, 31] for
standard references.

Note that this is similar to the situation we have in the context of Edwards’
speculation, where the zeros $t_{n}$ of the core $Z_{0}(t)$ are expected to be
rough initial guesses of the zeros of $Z(t)$, as discussed in Section 4. Thus,
it is more than natural to suggest the Newton method as a natural candidate
for the required method vaguely anticipated by Edwards that would

_’…allow one to go from a zero of the first term $Z_{0}(t)$ to a nearby zero
of $Z(t)$’._

However, if the Newton method or any other, similarly classical, numerical
method is indeed the method Edwards has in mind in his speculation, one comes
to wonder why does he not specify it directly but rather remain so vague
regarding its concrete definition?

In fact, not only does Edwards remain vague regarding the precise definition
of the method, after stating the speculation he actually goes to express
strong reservations regarding the possible universal applicability of any such
a method altogether. In particular, he draws parallels to Gram’s law, noting
scenarios where it could completely fail. For instance, putting aside the
challenges we described in Section 5, the following cautionary note might have
deterred further investigation into the relation between $Z_{0}(t)$ and
$Z(t)$:

_’The method can certainly fail. To see how completely it can fail, it
suffices to consider a complete failure of Gram’s law, for example, the
failure between $g_{6708}$ and $g_{6709}$ in Lehmer’s graph.’_ (ibid.)

Nevertheless, it is illuminating to investigate the Newton method as an
initial attempt to find the required method for transitioning from zeros of
$Z_{0}(t)$ to zeros of $Z(t)$. Of course, it should be noted that if the
Newton method, starting at $t_{n}\in\mathbb{R}$, would indeed converge to a
unique real zero $\widetilde{t}_{n}\in\mathbb{R}$ of $Z(t)$ for any
$n\in\mathbb{Z}$, it would imply the RH.

Let us also note that since our investigation in this section will be based
strictly on numerical examples, for reasons that will become clear shortly,
the theoretical reservations of Section 5 do not apply. Thus, we can employ
the classical Riemann-Siegel formula for this purpose. Consider the following
example of the zeros mentioned in Edwards’ concern:

######  Example 6.1 (The first Lehmer pair of zeros).

Let us apply Newton’s method for the two Lehmer zeros referred to by Edwards.
Starting from $t_{6708}$ and $t_{6709}$, The two corresponding zeros of $Z(t)$
are known to be given approximately by

$\begin{array}[]{ccc}\widetilde{t}_{6708}\approx
7005.06&;&\widetilde{t}_{6709}\approx 7005.10,\end{array}$ (16)

they are characterized as a pair of zeros of $Z(t)$ of unusual close proximity
to each other. The first Newton iterations $t^{k}_{6708}$ and $t^{k}_{6709}$
for these two zeros are presented in Table 1:

k | $t^{k}_{6708}$ | $t^{k}_{6709}$
---|---|---
1 | 7004.95 | 7005.84
2 | 7005.01 | 7005.23
3 | 7005.04 | 7005.15
4 | 7005.05 | 7005.12
5 | 7005.06 | 7005.10
Table 1. Newton’s Iterations for $t^{k}_{6708}$ and $t^{k}_{6709}$

The table demonstrates that, despite Edwards’ reservations about the possible
effect of the unusual proximity of the Lehmer zeros $t_{6708}$ and $t_{6709}$,
the iterations $t^{k}_{6708}$ and $t^{k}_{6709}$ nevertheless do converge to
$\widetilde{t}_{6708}$ and $\widetilde{t}_{6709}$ as given in (16),
respectively, without encountering any notable issues in practice.

In fact, experimentation shows that Newton’s method tends to work surprisingly
well for a vast number of zeros in a similar manner to that presented in the
above example. However, despite the apparent success of Newton’s method in
relating a substantial number of zeros $t_{n}$ of $Z_{0}(t)$ to corresponding
zeros $\widetilde{t}_{n}$ of $Z(t)$, our following example reveals that
Edwards’ original concerned intuition is far from being unjustified, and
Newton’s method does eventually fail, albeit for far more distant zeros than
the ones he originally proposed222Zeros such as $\widetilde{t}_{730120}$ and
$\widetilde{t}_{730121}$ were most certainly beyond the range of numerical
verification available to Edwards at the time. In fact, in [4], he reproduced
the graph of $Z(t)$ near the Lehmer zeros originally shown in [19]. Lehmer
notes that this computation required ”only” a few hours of machine time on a
SWAC computer at the United States National Bureau of Standards, used for this
purpose.:

######  Example 6.2 (Failure of Newton’s method for the $730120$-th zero).

Consider the $730120$-th zero of $Z(t)$, given by standard zero tables as
follows:

$\widetilde{t}_{730121}=450613.8005.$

Direct computation shows that if we initiate Newton’s method from the
$730120$-th and $730121$-th zeros of $Z_{0}(t)$, given by:

$\begin{array}[]{ccc}t_{730120}=450613.9649&;&t_{730121}=450614.5269,\end{array}$

both converge to the $730121$-th zero of $Z(t)$:

$lim_{k\rightarrow\infty}(t_{730120}^{k})=lim_{k\rightarrow\infty}(t_{730121}^{k})=\widetilde{t}_{730121}.$
(17)

This is visually illustrated in Fig. 3 which shows $\ln\lvert Z(t)\rvert$
(blue), $\ln\lvert Z_{0}(t)\rvert$ (orange) and the locations of the zeros
$t_{n-2}$, $t_{n-1}$ and $t_{n}$ of $Z_{0}(t)$ for $n=730121$ (purple) in the
region $450613.4\leq t\leq 450614.6$:

Figure 3. Graphs of $\ln\lvert Z(t)\rvert$ (blue), $\ln\lvert Z_{0}(t)\rvert$
(orange) and the zeros $t_{n-2}$, $t_{n-1}$ and $t_{n}$ of $Z_{0}(t)$ for
$n=730121$ (purple) in the region $450613.4\leq t\leq 450614.6$.

In the setting of Fig. 3, the zeros $t_{n}$ of $Z_{0}(t)$ (purple) can be
envisioned as balls positioned on the graph of $\ln|Z(t)|$ directly above
their corresponding locations on the $t$-axis. Newton’s method seeks to
identify the zero by effectively rolling these balls down the curve of
$\ln|Z(t)|$ until a zero is located (illustrated by black arrows). As
demonstrated in Fig. 3, this process results in both $t_{730120}$ and
$t_{730121}$ converging to $\widetilde{t}_{730121}$, while $t_{730119}$
converges to $\widetilde{t}_{730119}$, thereby failing to identify
$\widetilde{t}_{730120}$ in the middle.

In particular, the above example shows that direct, naive, application of
Newton’s method fails to locate certain zeros of $Z(t)$, and thus cannot
confirm their realness. It serves as a proof by counterexample of:

######  Corollary 6.1 (Failure of Newton’s method).

Newton’s method, starting from $t_{n}$, the $n$-th zero of the core
$Z_{0}(t)$, does not always converge to $\widetilde{t}_{n}$, the $n$-th real
zero of $Z(t)$.

So far, our analysis of Edwards’ speculation through classical methods
identifies two main crucial obstacles:

The bad:

The direct application of classical methods like Newton’s method is generally
ineffective and fails for the purpose of locating all zeros, due to their
insufficient sensitivity.

The ugly:

While the Riemann-Siegel formula is efficient for numerical calculations, for
developing theoretical analytical arguments that would be valid for all
$n\in\mathbb{Z}$, this formula is completely unfeasible due to the non-closed
form of the coefficients $b_{n}$ and $c_{n}$ involved in estimating the error
term $R(t)$.

These two obstacles demonstrate that, from a classical perspective, the
questions raised in Section 3 due to Edwards’ supposed vagueness indeed
originate from deep, non-trivial issues concerning the zeros of the
$Z$-function.

Hence, to find a well-defined method for relating the zeros of the core
$Z_{0}(t)$ to those of $Z(t)$, it becomes mandatory to propose an alternative
to the Riemann-Siegel formula. Additionally, this alternative should
facilitate a more nuanced procedure than the classical Newton’s method for
transitioning between zeros.

##  7\. Spira’s high-order sections vs. the Riemann Siegel Formula

Recall that sections of the Hardy $Z$-function are given by

$Z_{N}(t):=Z_{0}(t)+\sum_{k=1}^{N}\frac{\cos(\theta(t)-\ln(k+1)t)}{\sqrt{k+1}}$
(18)

for any $N\in\mathbb{N}$, see [15, 29, 30]. These sections are known to
approximate the Hardy $Z$-function in two distinct ways. The first, the
classical Hardy-Littlewood approximate functional equation (6), is expressed
as

$Z(t)=2Z_{\widetilde{N}(t)}(t)+O\left(\frac{1}{\sqrt{t}}\right),$ (19)

where $\widetilde{N}(t)=\left[\sqrt{\frac{t}{2\pi}}\right]$. The second, a
less conventional approximation to which we refer as Spira’s approximation, is

$Z(t)=Z_{N(t)}(t)+O\left(\frac{1}{\sqrt{t}}\right),$ (20)

with considerably more terms $N(t)=\left[\frac{t}{2}\right]$. We have explored
the remarkable properties of this Spira approximation (20) in [15],
demonstrating that unlike the classical Hardy-Littlewood AFE (6), which
necessitates further correction by the Riemann-Siegel formula, as discussed in
Section 2, the Spira approximation is sufficiently sensitive to discern the RH
without additional correction. In particular, based on our previous work on
approximate functional equations for series acceleration [14], we have proven:

######  Theorem 7.1 ([15]).

All the non-trivial zeros of the Hardy $Z$-function $Z(t)$ are real if and
only if all the zeros of the high-order Spira approximation $Z_{N(t)}(t)$ are
real.

For instance, Fig. 4 displays $\ln\left|Z(t)\right|$ (orange) alongside
Spira’s high-order approximation $\ln\left|Z_{N(t)}(t)\right|$ (blue) within
the range $412<t<419$:

Figure 4. Graphs of $ln\lvert Z(t)\rvert$ (orange) and Spira’s high-order
approximation $\ln\left|Z_{N(t)}(t)\right|$ (blue) within the range
$412<t<419$

Figure 4 shows that indeed Spira’s approximation captures the two real zeros
missed by the Hardy-Littlewood approximation of Fig 2.

######  Remark 7.2.

The fact that Spira’s approximation (20), whose error is
$O\left(\frac{1}{\sqrt{t}}\right)$, affords better approximation than the
Riemann-Siegel formula whose error bound is better than
$O\left(\frac{1}{\sqrt{t}}\right)$ even for the first term, might seem
counter-intuitive. Our result in [15] demonstrates that, in practice, Spira’s
approximation achieves further improved accuracy beyond the
$O\left(\frac{1}{\sqrt{t}}\right)$ bound obtained for it via classical
methods. In fact we show that via acceleration methods that, it converges, as
$t$ increases, to an, auxiliary, third approximation of exceptional level of
accuracy, $O\left(e^{-\omega\cdot t}\right)$, where $\omega>0$ is a specific
positive constant.

Our objective in the next section is to introduce a new variational method,
based on Spira’s approximation (20), aimed at overcoming the obstacles faced
by classical techniques of the Riemann-Siegel formula, described so far.

##  8\. The Variational Approach to Edwards’ Speculation

The unique relation between the zeros of $Z(t)$ and those of Spira’s
approximation $Z_{N(t)}$ expressed in Theorem 7.1 enables us to give Edwards’
speculation a mathematically well-defined interpretation, in the setting of
higher-order sections. Let us introduce the following central definition:

######  Definition 8.1 ($N$-th $A$-Parameter Space).

For a given $N\in\mathbb{N}$, we define $\mathcal{Z}_{N}$ to be the
$A$-parameter space consisting of functions of the form

$Z_{N}(t;\overline{a})=Z_{0}(t)+\sum_{k=1}^{N}\frac{a_{k}}{\sqrt{k+1}}\cos(\theta(t)-\ln(k+1)t),$

where $\overline{a}=(a_{1},\ldots,a_{N})$ belongs to $\mathbb{R}^{N}$.

In particular, within the $A$-parameter space $\mathcal{Z}_{N}$ we have two
unique elements of special importance:

  1. (1)

The core $Z_{0}(t)=Z_{N}(t;\overline{0})$, at the origin
$\overline{a}=\overline{0}$.

  2. (2)

The element $Z_{N}(t;\overline{1})$ at $\overline{a}=\overline{1}$, for which
$Z(t)\approx Z_{N}(t;\overline{1})$ in $2N\leq t\leq 2N+2$.

Edwards’ speculation now becomes a concrete question, that of studying the way
zeros of $Z_{N}(t;\overline{a})$, within the range $2N\leq t\leq 2N+2$, change
along continuous paths connecting $\overline{a}=\overline{0}$ to
$\overline{a}=\overline{1}$, attempting to find such paths which keep the
zeros real. Figure 5 shows a schematic illustration of this idea by presenting
the core $Z_{0}(t)=Z_{N}(t;\overline{0})$ connected to $Z_{N}(t;\overline{1})$
through two different curves in the $A$-parameter space $\mathcal{Z}_{N}$:

Figure 5. Schematic illustration of the core $Z_{0}(t)=Z_{N}(t;\overline{0})$
connected to $Z_{N}(t;\overline{1})$ through two different curves in the
$A$-parameter space $\mathcal{Z}_{N}$.

Despite the seeming simplicity of this new variational point-of-view to
Edwards’ speculation, the following should be noted:

  1. (1)

Within the framework of the classical Riemann-Siegel formula, it is completely
impossible to introduce a natural space of the form $\mathcal{Z}_{N}$, in
which both the core $Z_{0}(t)$ and $Z(t)$ are elements333 For simplicity, we
refer to $Z(t)$ as an element of $\mathcal{Z}_{N}$. By this, we specifically
mean that Spira’s section, $Z_{N}(t;\overline{1})$, is within
$\mathcal{Z}_{N}$ and serves, via a reformulation of (20), as a sufficiently
accurate approximation of $Z(t)$ in the relevant range $2N\leq t\leq 2N+2$,
for purposes of evaluating the RH. , for which Theorem 7.1 is instrumental.

Essentially, our space $\mathcal{Z}_{N}$ is derived by incorporating
parameters $a_{k}$ in front of the terms
$\frac{\cos(\theta(t)-\ln(k+1)t)}{\sqrt{k+1}}$ in the defining formula of
Spira’s approximation (20), and considering $N=N(t)$. Notably, all these terms
share a similar structure, naturally depending on the parameter $k$ in a
unified manner.

Conversely, within the Riemann-Siegel framework, there is no analogous method
for integrating such parameters, primarily because, as detailed in Section 5,
the components used to approximate the error term are not expressed in a
closed-form that is analytically manageable.

  2. (2)

Studying the change of given zeros of $Z_{N}(t;\overline{a})$ along continuous
paths in the parameter space of $\mathcal{Z}_{N}$ recasts the RH into a subtle
optimization problem, one that is inherently more sensitive than the Newton
method. Newton’s method, after all, is restricted to only one way for how
zeros of $Z_{0}(t)$ may converge to those of $Z(t)$. Specifically, within the
context of the $Z$-function, it tends to identify the closest zero of $Z(t)$
relative to the initially chosen zero $t_{n}$ of $Z_{0}(t)$. The allowance for
selecting a path introduces significant sensitivity and versatility. We will
further elaborate on why such an approach is particularly adept at properly
locating the zero of $Z(t)$ naturally corresponding to $t_{n}$, overcoming the
limitations identified for Newton’s method.

######  Remark 8.2.

The $A$-parameter space $\mathcal{Z}_{N}$, though not explicitly mentioned, is
implicit in [15]. In fact, the proof of Theorem 7.1 can be viewed as
demonstrating the existence of
$Z^{acc}_{N}(t)=Z_{N}(t;\overline{a}_{N}^{acc})$ within $\mathcal{Z}_{N}$,
referred to as the accelerated approximation satisfying
$Z(t)=Z^{acc}_{N(t)}(t)+O\left(e^{-\omega\cdot t}\right)$ as well as
$\|\overline{1}-\overline{a}^{acc}\|\rightarrow 0$ as $N\rightarrow\infty$,
with the distance defined by the natural Euclidean norm in $\mathcal{Z}_{N}$.

Consider a smooth parametrized curve $\gamma(r)$ in $\mathcal{Z}_{N}$ for
$r\in[0,1]$, connecting the core function $Z_{0}(t)$, at $r=0$, to
$Z_{N}(t;\overline{1})$, at $r=1$. Locally, for small enough $r>0$, every zero
$t_{n}$ of $Z_{0}(t)$ can be smoothly extended to a unique zero $t_{n}(r)$ of
$Z_{N}(t;\gamma(r))$. In general, one can always extend the definition of
$t_{n}(r)$ in a continuous, yet not necessarily smooth and unique manner444 In
analogy, consider the $1$-parametric family $\frac{1}{2}(1-t)(1+t)=r$ within
the space of quadrics. For $r=0$, there are two zeros $t_{\pm}=\pm 1$. For
$0\leq r<\frac{1}{2}$, one can uniquely and smoothly extend
$t_{\pm}(r)=\pm\sqrt{1-2r}$. However, at $r=\frac{1}{2}$, the two zeros
collide into a double zero at $t=0$, and subsequently turn into two conjugate
complex zeros. Specifically, for $\frac{1}{2}<r\leq 1$, the paths can be
continued in a continuous but non-smooth and non-unique manner, necessitating
a choice regarding which path proceeds to the upper half and which to the
lower half of the complex plane, beyond $r=\frac{1}{2}$. .

A priori, one might expect that the zeros $t_{n}(r)$ can move arbitrarily in
the complex plane, around $t_{n}$. The following theorem demonstrates that
there is a crucial restriction on the zero to, locally, remain real,
independent of the curve $\gamma(r)$:

######  Theorem 8.3 (Local realness of zeros).

Let $\gamma(r)$ be a smooth parametrized curve in $\mathcal{Z}_{N}$ for
$r\in[0,1]$, connecting the core function $Z_{0}(t)$, at $r=0$, to
$Z_{N}(t;\overline{1})$, at $r=1$. Then the zeros $t_{n}(r)$ of
$Z_{N}(t;\gamma(r))$ remain well-defined, smooth, and real as long as they do
not collide with other consecutive zeros, that is, as long as $t_{n}(r)\neq
t_{n\pm 1}(r)$.

###### Proof.

The Riemann-Siegel theta function satisfies
$\overline{\theta(\overline{t})}=\theta(t)$. Consequently, all terms of
$Z_{N}(t;\gamma(r))$ satisfy

$\overline{\cos(\theta(\overline{t})-\ln(k+1)\overline{t})}=\cos(\theta(t)-\ln(k+1)t).$

This implies that the set of zeros of $Z_{N}(t;\gamma(r))$ is self-conjugate
for any $r\in[0,1]$. That is, if $t\in\mathbb{C}$ is a zero of
$Z_{N}(t;\gamma(r))$, then its complex conjugate $\overline{t}$ is also a
zero. Given that $t_{n}(0)=t_{n}\in\mathbb{R}$ are all real and $t_{n}(r)$
varies continuously with $r$, zeros can only transition off the real line as
conjugate pairs in the event of a collision between adjacent zeros. ∎

Theorem 8.3 implies that, although elements of $\mathcal{Z}_{N}$ have an
infinite number of zeros, these zeros exhibit behaviour rather similar to that
of polynomials with real coefficients. In particular, Theorem 1 from the
introduction follows directly:

######  Corollary 8.4 (Edwards’ Speculation for High-Order Sections).

The Riemann Hypothesis holds if and only if, for any $n\in\mathbb{Z}$, there
exists a path $\gamma(r)$ in $\mathcal{Z}_{\left[\frac{t_{n}}{2}\right]}$ from
$\overline{a}=\overline{0}$ to $\overline{a}=\overline{1}$, along which
$t_{n}$ does not collide with its adjacent zeros $t_{n\pm 1}$. That is, for
which $t_{n}(r)\neq t_{n\pm 1}(r)$ for all $r\in[0,1]$.

In its essence, Corollary 8.4 offers a new point-of-view on the RH, exhibiting
it as a type of non-linear optimization problem. Indeed, it shows that RH
requires finding a curve $\gamma_{n}(r)$ connecting $Z_{0}(t)$ to $Z_{N}(t)$
in a non-colliding way, for any $n\in\mathbb{Z}$. Taking the analogy to
polynomials with real coefficients, we anticipate the existence of a
discriminant function $\Delta_{n}(\overline{a})$ on the space
$\mathcal{Z}_{N}$, such that the hyper-surface $\Delta_{n}(\overline{a})=0$ is
the closure of all instances such that $t_{n}(r)=t_{n+1}(r)$.

Hence, we are looking to effectively construct the curve $\gamma_{n}(r)$,
starting from the core $Z_{0}(t)$ and incrementally increasing $\overline{a}$
from $\overline{a}=\overline{0}$ to $\overline{a}=\overline{1}$ in a manner
that minimizes the decrease in $\Delta_{n}(\overline{a})$, avoiding collision,
that is vanishing of $\Delta_{n}(\overline{a})=0$. The main tool for solving
such problems is the Karush-Kuhn-Tucker optimization theorem, see [16, 17].
However, if one is willing to ease on optimality, analysis of
$\Delta_{n}(\overline{a})$ might lead to the description of curves
$\gamma_{n}(r)$, more natural than the optimal one. This exploration, would
require the formal definition and analysis of $\Delta_{n}(\overline{a})$,
which is reserved for future discourse.

##  9\. The $A$-variation method - Example for the $730120$-th Zero

In this section we describe the application of the $A$-variation method to the
specific case of the $730120$-th zero, for which the classical Newton method
fails, as shown in Section 6. While our focus here is on numerical examples,
the rationale behind their presentation is twofold. Firstly, they demonstrate
the $A$-variation method in action for a specific case, providing essential
illustration of the concepts discussed thus far. Secondly, we propose that the
principles outlined may reflect general, universal properties of the variation
of zeros of $Z(t)$, underscoring the necessity of more comprehensive
theoretical investigation.

###  9.1. The linear curve

The most natural curve to take in the space $\mathcal{Z}_{N(t)}$ to connect
$Z_{0}(t)$ to $Z_{N(t)}(t;\overline{1})$ is the linear curve given by

$Z_{N(t)}(t;\overline{r}):=Z_{0}(t)+\sum_{k=1}^{N(t)}\frac{r}{\sqrt{k+1}}cos(\theta(t)-ln(k+1)t),$
(21)

for $r\in[0,1]$, illustrated in purple in Fig. 5. We consider the dynamic
evolution of the functions $Z_{N(t)}(t;\overline{r})$ from $r=0$ to $r=1$ in
the range $450613.4\leq t\leq 450614.6$, within which we have the three zeros
$t_{n-2},t_{n-1},t_{n}$ of $Z_{0}(t)$ for $n=730121$. Given the varied
dynamics across different segments of $r\in[0,1]$, we will present each part
individually.

Step 1:

Figure 6 shows the graphs of $\ln\lvert Z_{N(t)}(t;r)\rvert$ for various
values within the initial interval $r\in[0,0.2425]$:

Figure 6. $\ln\lvert Z_{N(t)}(t;r)\rvert$ in $450613.4\leq t\leq 450614.6$ for
$r\in[0,0.2425]$.

Within the interval $r\in[0,0.2425]$ the two zeros $t_{730120}(r)$ and
$t_{730121}(r)$ are seen to proceed to collide one with the other, while
$t_{730119}(r)$ remains relatively static.

Step 2:

Figure 7 shows the graphs of $\ln\lvert Z_{N(t)}(t;r)\rvert$ for various
values within the interval $r\in[0.2425,0.85]$, together with $ln\lvert
Z_{0}(t)\rvert$ (blue):

Figure 7. $\ln\lvert Z_{N(t)}(t;r)\rvert$ in $450613.4\leq t\leq 450614.6$ for
$r\in[0.2425,0.85]$.

The figure shows the development of $\ln\lvert Z_{N(t)}(t;r)\rvert$ after the
collision of $t_{730120}(r)$ and $t_{730121}(r)$. In particular, in this
region the two corresponding zeros are not real and hence are not presented in
the figure, which shows the values only over the real $t$-axis. Indeed,
presenting the values $\ln\lvert Z_{N(t)}(t;r)\rvert$ in a three dimensional
plot over the complex plane would have shown that the two corresponding zeros
are now actually complex conjugate to each other, as guaranteed in the proof
of Theorem 8.3, proceeding to move together parallel down to the left of the
$t$-axis.

Step 3:

Figure 8 shows the graphs of $\ln\lvert Z_{N(t)}(t;r)\rvert$ for various
values within the interval $r\in[0.85,0.985]$:

Figure 8. $\ln\lvert Z_{N(t)}(t;r)\rvert$ in $450613.4\leq t\leq 450614.6$ for
$r\in[0.85,0.985]$.

While in this region the two zeros are still not real, they are starting their
return to the real $t$-axis, as the RH obliges.

Step 4:

Figure 9 shows the graphs of $\ln\lvert Z_{N(t)}(t;r)\rvert$ for various
values within the final interval $r\in[0.985,1]$:

Figure 9. $\ln\lvert Z_{N(t)}(t;r)\rvert$ in $450613.4\leq t\leq 450614.6$ for
$r\in[0.985,1]$.

In this region of $r\in[0.985,1]$ the two zeros are real and have returned to
the real $t$-axis, they advance to position themselves in their final location
as zeros of $Z(t)$, compare Fig. 3.

The above example of the linear curve $Z_{N(t)}(t;r)$ shows that it already
seems to mitigate a few of the problems encountered with the classical Newton
method. Indeed, the issue encountered in Example 6.2 was the convergence of
both zeros $t_{730120}$ and $t_{730121}$ of $Z_{0}(t)$ to a single zero
$\widetilde{t}_{730121}$ of $Z(t)$, missing $\widetilde{t}_{730120}$. In
contrast, tracking the dynamics of $Z_{N}(t;r)$, we can trace two continuous
paths $t_{730120}(r)$ and $t_{730121}(r)$, starting from $t_{730120}$ and
$t_{730121}$ at $r=0$ and ending at both $\widetilde{t}_{730120}$ and
$\widetilde{t}_{730121}$ for $r=1$, as required.

However, these paths are not unique due to collisions555In Fig. 5, these
collisions are schematically depicted as instances where the linear curve (in
purple) transitions from the blue region, where the zeros are real, to the red
region, where the zeros are complex conjugate, and vice versa. The non-
colliding curve is represented in cyan. occurring around $r=0.2425$ and
$r=0.985$. Given our pursuit for general theoretical principles rather than
empirical verification, such colliding paths introduce numerous disadvantages.
For instance, while it is feasible to numerically trace the paths of zeros
$t_{n}(r)$ in specific examples, the general scenario raises difficulties as
collisions could potentially lead to complex configurations, especially if a
pair of conjugate zeros meets another pair from a different collision,
complicating theoretical tracking.

Furthermore, and most critically for the RH, if one allows for collisions, it
is not clear what general mechanism should guarantee the return of the zeros
to the real line at a later time, as indeed observed to occur in the presented
example. In fact, it can be argued that theoretical identification of such a
mechanism is instrumental for understanding of the underlying origin of the
RH.

###  9.2. A non-colliding curve

Due to the limitations outlined above our approach is to try to replace the
linear curve by a more sensitive curve $\gamma_{n}(r)$ within the space
$\mathcal{Z}_{N}$, avoiding collisions altogether. This is where the
variational method shows its true power, as the space $\mathcal{Z}_{N}$ allows
for a vast option of curves through which to connect $Z_{0}(t)$ to
$Z_{N}(t;\overline{1})$, besides the linear curve.

In order to describe such a non-colliding curve, let us note that we observe
two distinct phenomena occurring in the dynamics of the zeros $t_{n}(r)$
presented for the linear curve. Firstly, the zeros draw nearer to each other
and, secondly, they shift together to the left along the $t$-axis. In
particular, we view the collision occurring due to the fact that the shift to
the left occurs in an uncorrelated manner with the tendency of the zeros to
come in proximity to one another. We thus aim to construct the non-colliding
curve by breaking the linear curve into two separate curves, each responsible
for its own task, to occur consecutively one after the other. The first is
responsible for achieving the shift to the left while minimizing the relative
change in distance between the two zeros, and the second leads to
$Z_{N(t)}(t;\overline{1})$, bringing the two zeros closer to their final
position.

We consider the following $2$-parametric system of sections arising from a
splitting of the indices (8.1):

$Z_{N}(t;r_{1},r_{2}):=Z_{0}(t)+\sum_{k\in
I_{shift}}\frac{r_{1}}{\sqrt{k+1}}cos(\theta(t)-ln(k+1)t)+\\\ +\sum_{k\in
I_{collide}}\frac{r_{2}}{\sqrt{k+1}}cos(\theta(t)-ln(k+1)t)\in\mathcal{Z}_{N},$
(22)

where we take the shifting indices to be

$I_{shift}=\left\\{1,2,4,6,12\right\\},$ (23)

and the colliding set of indices $I_{collide}$ is taken to be the complement
of $I_{shift}$. In this case, the split of indices into shifting and colliding
has been obtained experimentally.

Shifting stage:

Figure 10 shows graphs of $\ln\lvert Z_{N(t)}(t;r_{1},r_{2})\rvert$ for the
following values of $(r_{1},r_{2})$: $(0,0)$ (blue), $(0.25,0.05)$ (yellow),
$(0.55,0.15)$ (green), (0.75,0.25) (red), $(1,0.41)$ (purple) in the range
$450613.58\leq t\leq 450614.8$:

Figure 10. Graphs of $\ln\lvert Z_{N(t)}(t;r_{1},r_{2})\rvert$ along the
following values of $(r_{1},r_{2})$: $(0,0)$ (blue), $(0.25,0.05)$ (yellow),
$(0.55,0.15)$ (green), (0.75,0.25) (red), $(1,0.41)$ (purple) in the range
$450613.58\leq t\leq 450614.8$.

We see that as $(r_{1},r_{2})$ transition along the outlined curve from
$(0,0)$ (blue) to $(1,0.41)$ (purple) the zeros $t_{n}(r)$ and $t_{n-1}(r)$
transition continuously to the left in a way that generally preserves their
relative distance. In particular, contrary to the case of the linear curve,
this transition to the left is now organized in a way that avoids the un-
required collision.

Descending stage:

Figure 11 shows the graphs of $\ln\lvert Z_{N(t)}(t;r_{1},r_{2})\rvert$ along
the following points $(r_{1},r_{2})$: $(1,0.41)$ (blue), $(1,0.6)$ (yellow),
$(1,0.8)$ (green), (1,0.95) (red), $(1,1)$ (purple) in the range
$450613.58\leq t\leq 450614.8$:

Figure 11. Graphs of $\ln\lvert Z_{N(t)}(t;r_{1},r_{2})\rvert$ along the
following points $(r_{1},r_{2})$: $(1,0.41)$ (blue), $(1,0.6)$ (yellow),
$(1,0.8)$ (green), (1,0.95) (red), $(1,1)$ (purple) in the range
$450613.58\leq t\leq 450614.8$.

As required, collisions have been avoided altogether in the transition from
$Z_{0}(t)$ to $Z_{N}(t;\overline{1})$ achieved by concatenating the shifting
and descending stages described above. In particular, the paths of zeros
$t_{730120}(r)$ and $t_{730121}(r)$ are defined in a unique and continuous
way, starting from $t_{730120}$ and $t_{730121}$ and terminating at
$\widetilde{t}_{730120}(r)$ and $\widetilde{t}_{730121}(r)$, respectively, in
a way that identifies the realness of both zeros.

The overall process can be conceptualized by viewing the shifting indices as
utilizing their potential for a collision to instead shift the zeros to the
left, effectively depleting the energy required for further collisions. On the
other hand, in situations where the linear curve inherently avoids collisions,
the shifting component’s contribution tends to be minor.

In this example of $n=730121$, the split of indices into shifting and
descending indices has been mainly obtained experimentally, as mentioned.
Justifying the existence of a similar split for any $n\in\mathbb{Z}$, where
the linear curve requires correction, even if just through a heuristic
plausibility argument, would greatly enhance our understanding of the RH. This
may necessitate the discovery of new fundamental properties of the function
$Z(t)$. We propose that a broad theoretical exploration of such traits would
require in-depth study of the geometric aspects of the proposed discriminant
$\Delta_{n}(\overline{a})$.

######  Remark 9.1 (The $A$-variation method and Newton’s method).

In fact, the role of the Newton method is not entirely negated within the
$A$-variation method framework. In practice, to trace the path of zeros
$t_{n}(r)$, it is still necessary to segment $r\in[0,1]$ into smaller
intervals $0<r_{1}<r_{2}<\ldots<r_{m}=1$, and inductively find
$t_{n}(r_{j+1})$ as the nearest zero of $Z_{N(t)}(t;\gamma(r_{j+1}))$ to
$t_{n}(r_{j})$ via the classical Newton method. Thus, the failure of Newton’s
method, as discussed in Section 6, primarily concerns its direct application
from $Z_{0}(t)$ to $Z(t)$ in a single leap. The $A$-variation method, however,
allows to facilitate a connection from $Z_{0}(t)$ to $Z(t)$ through more
gradual steps, potentially ensuring that the transition at each stage uniquely
leads to real zeros.

##  10\. Summary and concluding remarks

The incremental advancements that have been achieved on the RH since its
introduction by Riemann in 1859 owe mainly to a tradition of scholarship based
on engagement with previous historical mathematical studies on the subject, as
well as the exploration of methods for the direct computation of zeta on the
critical line, both approaches leading back to Riemann’s original work itself.
For instance, the discovery of the Riemann-Siegel formula was a direct
consequence of Siegel’s thorough examination of Riemann’s original,
unpublished, computational manuscripts. Building on this tradition, Edwards’
speculation, through his detailed analysis of the inner-workings of the
Riemann-Siegel formula, reminiscent of Siegel’s approach to Riemann’s work,
inspired contemporary questions based on insights that may have been
envisioned by Riemann himself.

At the foundation of Edwards’ speculation is the notion that a conceptual,
mathematically well-defined link could be established between the zeros of the
core $Z_{0}(t)$ and those of $Z(t)$. However, following the introduction of
his speculation, Edwards turns to express strong reservations about the
practical application of such an approach, while offering only broad and
somewhat vague reasons for his doubts.

In the first part of this work, we have systematically analysed Edwards’
speculation, identifying in detail two main significant obstacles that hinder
its practical application within the original context of the Riemann-Siegel
formula. Firstly, we encounter the insurmountable challenge of accounting for
the error terms inherent in the Riemann-Siegel formula. Secondly, there is the
ambiguity surrounding the specific methodology required to effectively link
the core zeros with those of $Z(t)$. In the process we have showed the
inadequacy of methods such as the classical Newton’s method in this context.

In the second part of this work, we have adapted Edwards’ speculation to the
context of Spira’s high-order sections, a setting in which, remarkably, most
of the obstacles outlined in the first part for the Riemann-Siegel setting are
avoided. Spira conducted experimental studies on methods for the direct
computation of zeta on the critical line [29, 30]. In his studies, Spira
observed that sections $Z_{N}(t)$ of $Z(t)$ defined in (18) could approximate
$Z(t)$ in two distinct manners:

(1) Hardy-Littlewood AFE:

$Z(t)\approx 2Z_{\widetilde{N}(t)}(t)$ for
$\widetilde{N}(t)=\left[\sqrt{\frac{t}{2\pi}}\right]$.

(2) High-order approximation:

$Z(t)\approx Z_{N(t)}$ for $N(t)=\left[\frac{t}{2}\right]$.

The Hardy-Littlewood AFE is highly classical and lies in the basis of the
Riemann-Siegel formula. In comparison, the high-order approximation appears
inferior to the AFE in almost all aspects. It is significantly more
computationally demanding and, based on standard asymptotic analysis, expected
to be less efficient. Nonetheless, through his empirical investigations, Spira
suggested that, in contrast to the AFE, high-order approximations possess the
sensitivity required to observe the RH directly, eliminating the need for
further correction. In [15], we have theoretically validated Spira’s
observations by proving that the RH for Spira’s high-order sections is,
indeed, equivalent to the RH for $Z(t)$.

Based on these results, in the current work we have introduced the
$A$-variation space $\mathcal{Z}_{N}$ whose elements are given by

$Z_{N}(t;\overline{a})=Z_{0}(t)+\sum_{k=1}^{N}\frac{a_{k}}{\sqrt{k+1}}cos(\theta(t)-ln(k+1)t)$
(24)

where $\overline{a}=(a_{1},...,a_{N})\in\mathbb{R}^{N}$. This allowed us to
prove Theorem 1 which expresses Edwards’ speculation, and hence the RH, as an
optimization problem requiring the study of the way zeros of
$Z_{N(t)}(t;\overline{a})$ transition from $Z_{0}(t)$ to $Z_{N(t)}(t)$ along
various paths in $\mathcal{Z}_{N}$ from $\overline{a}=\overline{0}$ to
$\overline{a}=\overline{1}$, with the aim of finding such paths which avoid
collisions between consecutive zeros for any $n\in\mathbb{Z}$.

This new variational viewpoint on the RH leads to several immediate
foundational questions. Our initial analysis of the space $\mathcal{Z}_{N}$
reveals that the zeros of its elements, despite being infinite in number,
exhibit certain characteristics reminiscent of polynomials with real
coefficients. Intriguingly, our findings suggest that, akin to polynomials
with real coefficients, the space $\mathcal{Z}_{N}$ might also be endowed with
a well-defined discriminant hypersurface, with the RH deeply intertwined with
the geometry of this discriminant. This promising direction is something we
aim to delve into in future work.

It should be noted that our approach, while according to Edwards is deeply
rooted in Riemann’s own 19th-century investigations, carries a 21st-century
flair. Indeed, our strategy of tracing zeros along specific paths within the
vast-dimensional space $\mathcal{Z}_{N}$, coupled with our optimization
approach to the RH, shares similarities with non-linear optimization problems
encountered in modern applications. For instance, the use of gradient descent
techniques in high-dimensional spaces for training machine learning algorithms
[7]. This is not to imply that theoretical exploration of such ideas was
previously impossible. However, the capacity for practical investigation of
such large dimensional spaces and the non-linear optimization problems defined
therein was not readily accessible. For example, while Spira conducted
extensive and pioneering numerical investigations of sections of $Z(t)$ in the
60s and 70s, one might argue that technological limitations, including those
related to data visualization, may have prevented him from viewing the problem
through a variational lens, studying the zeros of sections $Z_{N}(t)$ for
given fixed $N\in\mathbb{N}$ as individual objects rather than their dynamic
change.

In fact, this intersection of our $A$-variation approach with techniques
intrinsic to machine learning not only underscores general theoretical
similarities but now also opens up new potential venues for studying the RH.
Indeed, our $A$-variation approach to the RH involves identifying non-
colliding paths in $\mathcal{Z}_{N}$ for any $n\in\mathbb{Z}$. In Section 9,
we successfully identified such a non-colliding curve for the specific example
of $n=730120$ through experimentation, pinpointing the shifting parameters by
direct methods. The exploration of $\mathcal{Z}_{N}$ with machine learning
algorithms to discover patterns that can facilitate the identification of such
non-colliding paths and their geometry in the general case heralds an
intriguing direction for future research. The potential application of these
techniques might represent a novel opportunity for contemporary examination of
the RH, creating a bridge between classical mathematical conjectures and
insights, following Riemann’s original explorations, to cutting-edge modern
optimization concepts.

## Declarations

### Funding

No funding was received to assist with the preparation of this manuscript.

### Conflicts of Interest/Competing Interests

The authors declare that they have no conflict of interest or competing
interests relevant to the content of this article.

### Data Availability

The authors declare that the data supporting the findings of this study are
available within the paper or from the corresponding author upon reasonable
request.

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