On a Generalized Moment Integral containing Riemann’s Zeta Function:

Analysis and Experiment
Michael Milgram,

Consulting Physicist, Geometrics Unlimited, Ltd.,

Box 1484, Deep River, Ont., Canada, K0J1P0.

<EMAIL_ADDRESS>
https://orcid.org/0000-0002-7987-0820
and
Roy J. Hughes,
School of Mathematics and Statistics,
University of New South Wales,
Kensington, 2033, Australia.
<EMAIL_ADDRESS>
https://orcid.org/0009-0009-1834-0111

MSC: 40G05, 40A10, 30B40, 11M06, 44A20
Keywords: Cesàro summation, improper integrals, auto-correlation, Riemann Zeta
function, discontinuous functions, quasi-periodic functions, Dirac comb
function

###### Abstract

Here, we study both analytically and numerically, an integral $Z(\sigma,r)$
related to the mean value of a generalized moment of Riemann’s zeta function.
Analytically, we predict finite, but discontinuous values and verify the
prediction numerically, employing a modified form of Cesàro summation.
Further, it is proven and verified numerically that for certain values of
$\sigma$, the derivative function $Z^{\prime}(\sigma,n)$ equates to one
generalized tine of the Dirac comb function without recourse to the use of
limits, test functions or distributions. A surprising outcome of the numerical
study arises from the observation that the proper integral form of the
derivative function is quasi-periodic, which in turn suggests a periodicity of
the integrand. This possibility is also explored and it is found
experimentally that zeta function values offset (shifted) over certain
segments of the imaginary complex number line are moderately auto-correlated.

##  1 Introduction

###  1.1 Background

In a previous work [1], relationships between different parametric instances
of inverse Mellin transform integrals of the form

$Z(\sigma,r)\equiv\int_{-\infty}^{\infty}\frac{\zeta(\sigma\pm
it)\,r^{\sigma\pm it}}{(\sigma\pm
it)}\,dt\,=-i\int_{\sigma-i\,\infty}^{\sigma+i\,\infty}{\zeta\left(v\right)r^{v-1}}dv,\hskip
20.0pt\sigma>1,$ (1.1)

were studied by the use of an extended form of Glasser’s Master Theorem [2]
for special values of the real variables $\sigma$ and $r$. In a second
previous work [3], similar integrals were studied by generalizing $r$ to
become a complex variable $r\rightarrow r\,\exp(i\phi)$, where it was found,
with $n\in\mathbb{N}$ and $r\in\Re$, that for certain values of $r=n$, the
value of the derivative function

$Z^{\prime}(\sigma,r)\equiv\frac{\partial}{\partial\,r}Z(\sigma,r)$ (1.2)

was indeterminate, depending on how the point $r=n$ was approached as a
function of $\phi$. In particular, it was found that if the point $r=n$ was
approached from certain directions in the complex $r$ plane, the function
$Z^{\prime}(\sigma,n)$ yielded a completely consistent family of finite
integrals. Approached from a different direction, the function
$Z^{\prime}(\sigma,n)$ diverged (i.e., was singular). Here we reconsider and
resolve that issue by first considering $Z(\sigma,r)$ as a function of $r$ and
show that it is discontinuous, that is

$\displaystyle
Z(\sigma,r)_{|_{\sigma>1}}=\int_{-\infty}^{\infty}\frac{\zeta\\!\left(\sigma+i\,t\right)r^{\sigma+i\,t}}{\sigma+i\,t}\,dt$
$\displaystyle=2\pi\lfloor r\rfloor,\hskip 20.0pt$
$\displaystyle\sigma>1,r\neq n,$ (1.3) $\displaystyle=\pi(2n-1),$
$\displaystyle\sigma>1,r=n.$ (1.4)

By differentiating the above with respect to $r$, it is clear that

$Z^{\prime}(\sigma,r\neq
n)_{|_{\sigma>1}}\equiv\frac{\partial}{\partial\,r}Z(\sigma,r)=\frac{1}{r}\,\int_{-\infty}^{\infty}\zeta\\!\left(\sigma+i\,t\right)r^{\sigma+i\,t}dt=0\,,\hskip
40.0pt\sigma>1,r\neq n$ (1.5)

and, by studying numerical approximations to the derivative function
$Z^{\prime}(\sigma,r=n)_{|_{\sigma>1}}$ at the discontinuity, we find that

$Z^{\prime}(\sigma,n)_{|_{\sigma>1}}=\int_{-\infty}^{\infty}\zeta\\!\left(\sigma+i\,t\right)n^{\sigma+i\,t}dt=\infty\,,\hskip
120.0pt\sigma>1\,.$ (1.6)

Combining (1.5) and (1.6) identifies, for any positive integer $n$,

$Z^{\prime}(\sigma,r)\equiv\int_{-\infty}^{\infty}\zeta\\!\left(\sigma+i\,t\right)r^{\sigma+i\,t}dt=2\pi\,r\,\delta(r-n)\,,\hskip
120.0pt\sigma>1\,,$ (1.7)

as one tine of the Dirac comb function when considered only as a function of
the variable $r$.111The Dirac comb function is the set of Dirac delta
functions of unit separation.

######  Remark 1.1.1.

For a proof of (1.7), see Section 3.1.

Since the main interest in such integrals focusses on the range $0<\sigma<1$,
in this work we consider that region by applying analytic continuation from
the region $\sigma>1$. Because $\zeta(\sigma+it)$ varies in sign throughout
its range and $|\zeta(\sigma+it)|/\log(t)$ is bounded when $\sigma>1,~{}t>2$,
[4, Corollary 1, page 184], it is reasonable to expect integrals of the form
(1.1) to be numerically convergent when $\sigma>1$. However, representations
with $\sigma<1$, which are not expected to always converge numerically, are
traditionally defined and given meaning by analytic continuation.

First, we study the special case $\sigma=3/2$ and then employ straightforward
translation of contour integrals (or the Master Theorem equivalent) to obtain
identities valid for $\sigma=1/2$. This will be found in Section 2. In Section
3, we then examine cases with general values of $\sigma$ following the same
methods. Acknowledging that we are dealing with functions on the very edge of
tractability both in a numerical and analytic sense, the following Section 4
applies the Cesàro regularization technique (see Appendix A) to ascertain if
the analytic results are consistent with numerical approximation.
Surprisingly, they are in excellent accord, given that the functions are
discontinuous (see Remark 2.2.2). Therefore we are reasonably confident that
our analytic results are valid - the numbers do not lie. An unexpected
digression arises when the various Cesàro approximations are viewed
graphically, since they suggest that proper integrals associated with (1.7)
are periodic, as are the integrands themselves. This observation is studied in
a further series of numerical experiments, showing in Section 5, that the
integrand function $|\zeta(\sigma+it)|$ is moderately auto-correlated .
Finally, our discoveries are summarized and discussed along with suggested
generalizations and applications.

###  1.2 Notation and Lemmas

####  1.2.1 Notation

Throughout the following, $0<r\in\Re$, $n,m\in\mathbb{N}$ always and $:=$
means symbolic replacement. Other symbols are real except if specified. We
employ $\lfloor...\rfloor$ and $\lceil...\rceil$ to represent the floor and
ceiling functions respectively and $\delta(x)$ is the Dirac delta function.
The $m^{th}$ derivative of $\zeta(s)$ is written $\zeta^{(m)}(s)$. Both
computer programs Maple [5] and Mathematica [6] were used throughout and are
individually cited where necessary.

######  Remark 1.2.1.

It is important to emphasize that, if $x\in\Re$, here we define floor and
ceiling functions such that

$\lfloor x\rfloor\text{ means the greatest integer less than {\bf but not
equal to} }x,$

and

$\lceil x\rceil\text{ means the smallest integer greater than {\bf but not
equal to} }x,$

as opposed to the common (e.g., Maple, Mathematica) usage $\lfloor n\rfloor=n$
and $\lceil n\rceil=n$. In other words, the floor and ceiling functions are
open at their respective ends. This means that the limit endpoints of $\lfloor
x\rfloor$ and $\lceil x\rceil$ are undefined and any identity containing these
functions requires that the values of that identity must be carefully
specified when $x=n$ if the identity is to be complete (e.g. [7, Eqs. II.1(3),
Eqs. II.1(4) and II.5(16)]. This is a consequence of, and flows naturally
from, the forthcoming analysis, where we independently obtain the value of a
function at a point of discontinuity, but not necessarily as the mean of the
limit of its values as the discontinuity is approached from above and below.

####  1.2.2 Lemmas - specific to $\sigma=3/2$

From [8, Eqs. 3.723(2) and 3.723(4)] with $\,r\,\in\Re$ and $j\in\mathcal{N}$,
we have

$\displaystyle\int_{-\infty}^{\infty}\frac{\sin\\!\left(t\,\ln\\!\left(\frac{j}{\,r\,}\right)\right)t}{t^{2}+\frac{9}{4}}dt$
$\displaystyle=\;{\pi}{\left(\frac{j}{\,r\,}\right)^{-\frac{3}{2}}},\hskip
20.0pt$ $\displaystyle\mathrm{if}~{}j>\,r\,;$
$\displaystyle=\,-{\pi}{\left(\frac{j}{\,r\,}\right)^{\frac{3}{2}}},\hskip
20.0pt$ $\displaystyle\mathrm{if}~{}j<\,r\,;$ $\displaystyle=\;0\hskip 20.0pt$
$\displaystyle\mathrm{if}~{}j=\,r\,,r=n,$ (1.8)

and

$\displaystyle\int_{-\infty}^{\infty}\frac{\cos\\!\left(t\,\ln\\!\left(\frac{j}{\,r\,}\right)\right)}{t^{2}+\frac{9}{4}}dt$
$\displaystyle=\frac{2}{3}\,\pi\left(\frac{j}{\,r\,}\right)^{-\frac{3}{2}}\hskip
20.0pt\mathrm{if}~{}j>\,r\,;$
$\displaystyle=\frac{2}{3}{\pi}{\left(\frac{j}{\,r\,}\right)^{\frac{3}{2}}},\hskip
20.0pt\mathrm{if}~{}j<\,r\,;$ $\displaystyle=\frac{2}{3}{\pi}\hskip
20.0pt\mathrm{if}~{}j=\,r\,,\,r\,\in\mathcal{N}\,.$ (1.9)

####  1.2.3 Lemmas - the general form

From the same source, more general forms of the same listed identities are

$\int_{-\infty}^{\infty}\frac{t\,\sin\\!\left(a\,t\right)}{\sigma^{2}+t^{2}}\,dt=\pi\,{\mathrm{e}}^{-a\,\sigma},\hskip
20.0pta>0,$ (1.10)

and

$\int_{-\infty}^{\infty}\frac{\cos\\!\left(a\,t\right)}{\sigma^{2}+t^{2}}\,dt=\frac{\pi}{\sigma}\,{\mathrm{e}}^{-a\,\sigma},\hskip
20.0pta>0.$ (1.11)

##  2 The special cases $\sigma=3/2$ and $\sigma=1/2$

###  2.1 The Master Theorem

Consider the function $F(t)$ defined, for $\,r\,>0$, $\,r\,\in\Re$, by

$F(t)\equiv\frac{\zeta\\!\left(\frac{1}{2}+i\,t\right)\,r\,^{\frac{1}{2}+i\,t}}{\frac{1}{2}+i\,t}-\frac{\zeta\\!\left(\frac{3}{2}-i\,t\right)\,r\,^{\frac{3}{2}-i\,t}}{\frac{3}{2}-i\,t}.$
(2.1)

It is easy to show that $F(t)+F(-i-t)=0$. Hence, from Glasser’s Master theorem
[2]

$\int_{-\infty}^{\infty}\left(\frac{\zeta\\!\left(\frac{1}{2}+i\,t\right)\,r\,^{\frac{1}{2}+i\,t}}{\frac{1}{2}+i\,t}-\frac{\zeta\\!\left(\frac{3}{2}-i\,t\right)\,r\,^{\frac{3}{2}-i\,t}}{\frac{3}{2}-i\,t}\right)dt=-2\,\pi\,\,r\,$
(2.2)

because only the residue at the point $t=-i/2$ contributes.

###  2.2 Evaluation: $\sigma=3/2$

We begin by considering the convergent integral

$\displaystyle Z(3/2,r)$
$\displaystyle\equiv\,\int_{-\infty}^{\infty}\frac{\zeta\\!\left(\frac{3}{2}-i\,t\right)\,r\,^{\frac{3}{2}-i\,t}}{\frac{3}{2}-i\,t}dt$
$\displaystyle=\mathop{\kern 0.0pt\mathop{\kern
0.0pt\sum}\limits_{j=1}}\limits^{\infty}\;{\left(\frac{j}{\,r\,}\right)^{-\frac{3}{2}}}\int_{-\infty}^{\infty}{\mathrm{e}}^{i\,t\,\ln\left(\frac{j}{\,r\,}\right)}\left(\frac{3}{2\,t^{2}+\frac{9}{2}}+\frac{i\,t}{t^{2}+\frac{9}{4}}\right)dt$
(2.3)

by writing

$\zeta\\!\left({3}/{2}-i\,t\right)=\mathop{\kern 0.0pt\mathop{\kern
0.0pt\sum}\limits_{j=1}}\limits^{\infty}\\!\frac{1}{j^{\frac{3}{2}-i\,t}}$
(2.4)

and noting that the summation and integration can be transposed because both
are convergent. When decomposed into its real and imaginary parts, we arrive
at

$\displaystyle\int_{-\infty}^{\infty}{\mathrm{e}}^{i\,t\,\ln\left({j}/{\,r\,}\right)}$
$\displaystyle\left(\frac{3}{2\,t^{2}+\frac{9}{2}}+\frac{i\,t}{t^{2}+\frac{9}{4}}\right)dt$
$\displaystyle=\int_{-\infty}^{\infty}\left(\left(\frac{3}{2}\right)\frac{\cos\\!\left(t\,\ln\\!\left(\frac{j}{\,r\,}\right)\right)}{t^{2}+\frac{9}{4}}-\frac{t\,\sin\\!\left(t\,\ln\\!\left(\frac{j}{\,r\,}\right)\right)}{t^{2}+\frac{9}{4}}\right)dt$
(2.5)

noting that the imaginary parts of the integral vanish by antisymmetry over
the integration range.

####  2.2.1 Case: $\,r\,=n$

In the case that $\,r\,$ is a positive integer $\,r\,=n$, we employ (1.8) and
(1.9) to find

$\displaystyle Z(3/2,n)$
$\displaystyle=\int_{-\infty}^{\infty}\frac{\zeta\\!\left(\frac{3}{2}-i\,t\right)n^{\frac{3}{2}-i\,t}}{\frac{3}{2}-i\,t}dt$
$\displaystyle=\pi\,\mathop{\kern 0.0pt\mathop{\kern
0.0pt\sum}\limits_{j=1}}\limits^{n-1}\\!\left(\frac{j}{n}\right)^{-\frac{3}{2}}\left({\left(\frac{j}{n}\right)^{\frac{3}{2}}}+{\left(\frac{j}{n}\right)^{\frac{3}{2}}}\right)+\mathop{\kern
0.0pt\mathop{\kern
0.0pt\sum}\limits_{j=n}}\limits^{n}\\!\frac{n^{3}\,\pi}{j^{3}}+\pi\,\mathop{\kern
0.0pt\mathop{\kern
0.0pt\sum}\limits_{j=n+1}}\limits^{\infty}\\!\frac{{\left(\frac{j}{n}\right)^{-\frac{3}{2}}}-{\left(\frac{j}{n}\right)^{-\frac{3}{2}}}}{\left(\frac{j}{n}\right)^{\frac{3}{2}}}$
$\displaystyle=\pi\,\mathop{\kern 0.0pt\mathop{\kern
0.0pt\sum}\limits_{j=1}}\limits^{n-1}\;2+\pi+0$
$\displaystyle=\pi\,\left(2\,n-1\right).$ (2.6)

####  2.2.2 Case $\,r\,\neq n$

In the case that $\,r\,>0$ is not a positive integer, that is $\,r\,\neq n$,
we again employ (1.8) and (1.9) to find

$\displaystyle Z(3/2,r)$
$\displaystyle=\int_{-\infty}^{\infty}\frac{\zeta\\!\left(\frac{3}{2}-i\,t\right)\,r\,^{\frac{3}{2}-i\,t}}{\frac{3}{2}-i\,t}dt$
$\displaystyle=\pi\,\mathop{\kern 0.0pt\mathop{\kern
0.0pt\sum}\limits_{j=1}}\limits^{{\lfloor\,r\,\rfloor}}\\!\left(\frac{j}{\,r\,}\right)^{-\frac{3}{2}}\left({\left(\frac{j}{\,r\,}\right)^{\frac{3}{2}}}+{\left(\frac{j}{\,r\,}\right)^{\frac{3}{2}}}\right)+\pi\,\mathop{\kern
0.0pt\mathop{\kern
0.0pt\sum}\limits_{j={\lceil\,r\,\rceil}}}\limits^{\infty}\\!\frac{{\left(\frac{j}{\,r\,}\right)^{-\frac{3}{2}}}-{\left(\frac{j}{\,r\,}\right)^{-\frac{3}{2}}}}{\left(\frac{j}{\,r\,}\right)^{\frac{3}{2}}}$
$\displaystyle=\pi\,\mathop{\kern 0.0pt\mathop{\kern
0.0pt\sum}\limits_{j=1}}\limits^{{\lfloor\,r\,\rfloor}}2+0$
$\displaystyle=2\,\pi\,\lfloor\,r\,\rfloor\,.$ (2.7)

Figure 1 shows both the analytic results obtained above, as well as a few
evaluations of Z(3/2,r) scattered over different values of $r$ obtained by
(difficult) numerical integration, demonstrating substantial agreement.

Figure 1: The staircase function $Z(\frac{3}{2},r)$ over a small range of $r$,
compared analytically and numerically. Note that the left and right limits of
each of the horizontal “treads” are open and that the values at the midpoints
of the “risers” are obtained by both an analytic and a numerical evaluation of
the integral, not by decree.

######  Remark 2.2.1.

Note that (2.7) does not reduce to (2.6) if $r\rightarrow n$ (see Remark
1.2.1) and therefore $Z(3/2,r)$ is discontinuous as a function of $r$ at
$r=n$. However, at $r=n$, the function $Z(3/2,n)$ does lie exactly at the
midpoint of the discontinuity, both analytically and numerically, as expected
[7, Eq. II.1(page 8)].

######  Remark 2.2.2.

As alluded to previously, the derivations presented here deal with subtle
issues of limits and discontinuities. To emphasize this point, Appendix B
presents homologous, but incorrect, identities obtained by a false derivation.
The resolution is left as a challenge for the reader.

###  2.3 Analytic continuation to $\sigma=1/2$

As noted in [1], by pairing integrals living inside the critical strip
$0\leq\sigma\leq 1$ with companions that are tractable and live outside, it
becomes possible to evaluate the companion integral $Z(1/2,r)$ by applying
either the Master Theorem or analytic continuation and compare with previous
results. From (2.2),(2.6) and (2.7) we find

$\displaystyle
Z(1/2,r)\equiv\int_{-\infty}^{\infty}\frac{\zeta\\!\left(\frac{1}{2}+i\,t\right)\,r\,^{\frac{1}{2}+i\,t}}{\frac{1}{2}+i\,t}dt$
$\displaystyle=\,2\,\pi\left({\lfloor\,r\,\rfloor-r}\right)\,,\hskip 30.0pt$
$\displaystyle r\neq n,$ (2.8) $\displaystyle=-\pi\,,$ $\displaystyle r=n.$
(2.9)

so that, when $n=2$, we have,

$\int_{-\infty}^{\infty}\frac{\zeta\\!\left(\frac{1}{2}+i\,t\right)2^{i\,t}}{\frac{1}{2}+i\,t}dt=-\frac{\pi}{\sqrt{2}}\,$
(2.10)

in agreement with [1, Eq. (4.13)]. Also, if $n=2$ we find

$\int_{-\infty}^{\infty}\frac{\zeta\\!\left(\frac{3}{2}-i\,t\right)2^{-i\,t}}{\frac{3}{2}-i\,t}dt=\frac{3\,\pi\,\sqrt{2}}{4},$
(2.11)

in agreement with [1, Eq.(4.12)].

###  2.4 The derivative

Figure 2: The function $A(r)$ over a small range of $r$.

For typographical clarity, define the function

$\displaystyle A(r)$ $\displaystyle=2\,\pi(\lfloor r\rfloor-r),\hskip
30.0pt~{}$ $\displaystyle r\neq n,$ (2.12) $\displaystyle=\,-\,\pi,$
$\displaystyle r=n,$ (2.13)

illustrated in Figure 2, demonstrating that $A(r)$ is invariant if
$\,r\,\rightarrow\,r\,+m$ and that it can be characterized as a sawtooth
function of period one and magnitude $2\pi$ where the special case $r=n$
corresponds to the midpoint of the sawtooth discontinuity. We also note that
the slope of $A(r)$ is continuous as $\,r\,\rightarrow n^{\pm}$, although the
function itself is discontinuous, so it is relevant to query the value of the
derivative $A^{\prime}(r)$ at $r=n$. By one simple, and standard definition,

$\displaystyle
A^{\prime}(n)\equiv\,A^{\prime}(r)\large|_{r=n}\equiv\underset{r\rightarrow
n}{lim}\,\frac{d}{d\,r}A(r)$ $\displaystyle=\underset{r\rightarrow
n}{lim}\,{\underset{h\rightarrow 0}{lim}}\,\frac{A(r+h)-A(r)}{h}$
$\displaystyle=2\pi\,\underset{r\rightarrow n}{lim}\,{\underset{h\rightarrow
0}{lim}}\,\frac{\lfloor r+h\rfloor-r-h-\lfloor r\rfloor+r}{h}$
$\displaystyle=2\pi\,\underset{r\rightarrow n}{lim}\,{\underset{h\rightarrow
0}{lim}}\,\frac{\lfloor r\rfloor-h-\lfloor r\rfloor}{h}$
$\displaystyle=-2\pi\,$ (2.14)

and similarly if $h$ is replaced by $-h$, so the derivative $A^{\prime}(n)$
exists everywhere and is continuous according to this definition. However, by
a second simple and standard definition

$\displaystyle A^{\prime}(n)\equiv\,A^{\prime}(r)\large|_{r=n}$
$\displaystyle={\underset{h\rightarrow 0}{lim}}\,\frac{A(n+h)-A(n)}{h}$
$\displaystyle=2\pi\,{\underset{h\rightarrow 0}{lim}}\,\frac{\lfloor
n+h\rfloor-n-h+\pi/2}{h}$ $\displaystyle=2\pi\,{\underset{h\rightarrow
0}{lim}}\,\frac{\lfloor n\rfloor-n-h+\pi/2}{h}$ $\displaystyle=\infty$ (2.15)

and the derivative is indeterminate when $r=n$, irrespective of our
requirement that $\lfloor n\rfloor$ is undefined.

The important point here is that $A(r)$ represents the function $Z(1/2,r)$
(see (2.9)) where the $r$ dependence only appears as an integrand term of the
form $r^{1/2+it}$ on the left-hand side for which the derivative is well-
known, continuous and consistent, independently of which of the two
definitions are employed. Hence we have uncovered a pathology where the left-
hand side of an identity always appears to be well-defined, and the right-hand
side sometimes is not. Similar indeterminacies have been observed elsewhere
[3] in related integrals, where it was speculated that there exists an
associated, uncategorized, possibly essential, singularity at $r=n$, when the
integral is studied as a function of complex $r$.

###  2.5 Differentiating

Differentiating (2.9) with respect to $r$ utilizing the definition (2.14),
yields

$\int_{-\infty}^{\infty}\zeta\\!\left({1}/{2}+i\,t\right)\,r\,^{i\,t}dt=-2\,\pi\,\sqrt{\,r}\,,\hskip
30.0ptr\neq n,$ (2.16)

because

$\frac{d}{d\,r\,}\lfloor\,r\,\rfloor=0\,,\hskip 100.0ptr\neq n.$ (2.17)

Including the condition, (2.16) agrees with [3, Eq. (5.28)], which was
obtained by an inverse Mellin transform, and reads

$\int_{0}^{\infty}\Re\left(\zeta\\!\left({1}/{2}+i\,t\right)\,r\,^{i\,t}\right)dt=-\pi\,\sqrt{\,r\,}\,,\hskip
30.0ptr\neq n\,.$ (2.18)

In the case that $r=n$, we also reproduce the indeterminacy alluded to above
and first observed in [3, Section (7.1)]. Utilizing (2.14) where
$A^{\prime}(r)$ is continuous when $r\rightarrow n^{\pm}$ and no condition
exists, by rewriting (2.18) as

$\int_{0}^{\infty}\left(\zeta_{I}\\!\left({1}/{2}+i\,v\right)\,\sin\\!\left(v\,\ln\\!\left(r\right)\right)-\zeta_{R}\\!\left({1}/{2}+i\,v\right)\,\cos\\!\left(v\,\ln\\!\left(r\right)\right)\right)dv=\pi\,\sqrt{r}$
(2.19)

we reproduce [3, Eq.(7.9)] when $r=n$; if we define the derivative
$\frac{d}{dr}A(r)$ to be infinite at the point $r=n$ as in (2.15), then we
reproduce the indefinite value obtained in [3, Eq.(7.10)] for that same
integral and the right-hand side of (2.19) becomes indefinite. We reiterate
that in this study we are considering a directed limit $r\rightarrow n^{\pm}$
for real values of $r$, in contrast to [3] where the limit $r\rightarrow n$
was a directional limit for complex values of $r$. In [3], it was shown that
the finite choice represented by (2.19) (and accordingly (2.14)), leads to a
consistent set of valuations for other integrals similar to (2.19). Finally if
$\,r\,=1$, (2.19) demonstrates that

$\int_{0}^{\infty}\zeta_{R}\\!\left({1}/{2}+i\,t\right)dt=-\pi\,,$ (2.20)

in agreement with [3, Eq. (5.26)], although it is unlikely that (2.20) is
numerically convergent and thereby it establishes one possible regularization
of the integral among an infinite number of possibilites.

##  3 The General Case

###  3.1 $\sigma>1$

As before, we begin by considering the convergent representation

$\zeta\\!\left(\sigma+i\,t\right)=\mathop{\kern 0.0pt\mathop{\kern
0.0pt\sum}\limits_{j=1}}\limits^{\infty}\\!\frac{1}{j^{\sigma}}\;{\mathrm{e}}^{-i\,t\,\ln\left(j\right)},\hskip
20.0pt\sigma>1,$ (3.1)

to be employed in the integrand of a convergent representation of
$Z(\sigma,r)$, and then, in Section 3.2, progress by analytic continuation to
the region $\sigma<1$, where both the integral and/or the sum may not
converge. Thus, with $\sigma>1$,

$Z(\sigma,r)\equiv\int_{-\infty}^{\infty}\frac{\zeta\\!\left(\sigma+i\,t\right)r^{\sigma+i\,t}}{\sigma+i\,t}\,dt=r^{\sigma}\mathop{\kern
0.0pt\mathop{\kern
0.0pt\sum}\limits_{j=1}}\limits^{\infty}\\!\frac{1}{j^{\sigma}}\int_{-\infty}^{\infty}\left(\frac{\sigma\,\cos\left(t\,\ln\left(\frac{r}{j}\right)\right)}{\sigma^{2}+t^{2}}+\frac{\sin\left(t\,\ln\left(\frac{r}{j}\right)\right)t}{\sigma^{2}+t^{2}}\right)dt\,,$
(3.2)

where the imaginary components vanish due to asymmetry. From (1.10) it follows
that

$\displaystyle\int_{-\infty}^{\infty}\frac{\sin\\!\left(t\,\ln\\!\left({r}/{j}\right)\right)}{\sigma^{2}+t^{2}}\,t\,dt$
$\displaystyle=\pi\left(\frac{r}{j}\right)^{-\sigma},\hskip 20.0pt$
$\displaystyle j<r;$ (3.3)
$\displaystyle=\,-\pi\left(\frac{r}{j}\right)^{\sigma},$ $\displaystyle j>r;$
(3.4) $\displaystyle=0,$ $\displaystyle j=r.$ (3.5)

Similarly, from (1.11)

$\displaystyle\sigma\int_{-\infty}^{\infty}\frac{\cos\\!\left(t\,\ln\\!\left(\frac{r}{j}\right)\right)}{\sigma^{2}+t^{2}}dt$
$\displaystyle=\pi\left(\frac{r}{j}\right)^{-\sigma},\hskip 20.0pt$
$\displaystyle j<r,$ (3.6)
$\displaystyle=\pi\left(\frac{r}{j}\right)^{\sigma},$ $\displaystyle j>r,$
(3.7) $\displaystyle=\pi,$ $\displaystyle j=r.$ (3.8)

####  3.1.1 $r\neq n$

If $r\neq n$, we find, as before (see (2.7))

$Z(\sigma,r)=\int_{-\infty}^{\infty}\frac{\zeta\\!\left(\sigma+i\,t\right)r^{\sigma+i\,t}}{\sigma+i\,t}\,dt=2\pi\lfloor
r\rfloor,\hskip 20.0pt\sigma>1.$ (3.9)

Since $\sigma$ is now a continuous variable, we can consider differentiating
(3.9) and (3.14) with respect to both $r$ and $\sigma$. First, differentiating
with respect to $\sigma$ (or integrating by parts), immediately yields the
identity

$\int_{-\infty}^{\infty}\frac{\zeta^{\left(1\right)}\\!\left(\sigma+i\,t\right)r^{\sigma+i\,t}}{\sigma+i\,t}dt=\int_{-\infty}^{\infty}\frac{\zeta\\!\left(\sigma+i\,t\right)r^{\sigma+i\,t}}{\left(\sigma+i\,t\right)^{2}}dt-2\,\pi\,{\lfloor
r\rfloor}\,\ln\\!\left(r\right)$ (3.10)

followed by differentiation with respect to $r$ which informs us that

$\int_{-\infty}^{\infty}\zeta^{\left(1\right)}\\!\left(\sigma+i\,t\right)r^{\sigma+i\,t}dt=0$
(3.11)

because $\frac{d{\lfloor r\rfloor}}{dr}=0$ if $r\neq n$. The generalization is
obvious:

$\int_{-\infty}^{\infty}\zeta^{\left(m\right)}\\!\left(\sigma+i\,t\right)r^{\sigma+i\,t}dt=0$
(3.12)

where $m\in\mathbb{N}$. Furthermore, differentiating (3.9) with respect to $r$
predicts that

$\frac{1}{r}\,\int_{-\infty}^{\infty}\zeta\\!\left(\sigma+i\,t\right)r^{\sigma+i\,t}dt=0\,,\hskip
60.0pt\sigma>1,~{}r\neq n,$ (3.13)

raising a question about the potential interpretation of (3.13) when $r=n$,
where the definition of “derivative” is ambiguous. See Section 4 below.

####  3.1.2 $r=n$

Exactly as in (2.6), we have

$Z(\sigma,n)=\int_{-\infty}^{\infty}\frac{\zeta\\!\left(\sigma+i\,t\right)n^{\sigma+i\,t}}{\sigma+i\,t}\,dt=\pi\left(2\,n-1\right),\hskip
20.0pt\sigma>1,$ (3.14)

and again, differentiating with respect to $\sigma$, gives

$\int_{-\infty}^{\infty}\frac{\zeta^{\left(1\right)}\\!\left(\sigma+i\,t\right)n^{\sigma+i\,t}}{\sigma+i\,t}dt=\int_{-\infty}^{\infty}\frac{\zeta\\!\left(\sigma+i\,t\right)n^{\sigma+i\,t}}{\left(\sigma+i\,t\right)^{2}}\,dt-\pi\left(2\,n-1\right)\,\ln\\!\left(n\right)\,.$
(3.15)

Notice that the case $n=1$ applied to (3.15) produces the identity

$\int_{-\infty}^{\infty}\frac{\zeta^{\left(1\right)}\\!\left(\sigma+i\,t\right)}{\sigma+i\,t}dt=\int_{-\infty}^{\infty}\frac{\zeta\\!\left(\sigma+i\,t\right)}{\left(\sigma+i\,t\right)^{2}}\,dt,\hskip
30.0pt\sigma>1.$ (3.16)

If (3.16) is compared to (3.10), it suggests that the (questionable) limit
$r\rightarrow 1$ applied to (3.10), is valid.

####  3.1.3 A Proof of (1.7)

When $\sigma>1$, in Section 4 it will be shown that the integral (3.13)
appears to be (numerically) divergent only if $r=n$, and since it vanishes
otherwise, we can justifiably write

$Z^{\prime}(\sigma,r)\equiv\,\int_{-\infty}^{\infty}\zeta\\!\left(\sigma+i\,t\right)r^{\sigma+i\,t}dt=2\pi\,r\,\delta(r-n)\,,\hskip
80.0pt\sigma>1\,,n\neq 0.$ (3.17)

The question that then arises is: “Is $2\pi$ the appropriate normalization?”

######  Remark 3.1.1.

It is important to understand that (3.17) differs from common invocations of
the Dirac delta function (for examples, see [7, Chapter V]) because it does
not involve limits, test functions or distributions. The approach to the limit
$r\rightarrow n$ is continuous, open-ended and always vanishes. In
discontinuous fashion, only at the limit point $r=n$ does the integral
diverge, and this appears to be numerically true.

######  Theorem 3.1.2.

$\int_{-\infty}^{\infty}\zeta\\!\left(\sigma+i\,v\right)r^{\sigma+i\,v}dv=2\pi\,r\,\delta(r-n)\,,\hskip
120.0pt\sigma>1\,,n\in\mathbb{N},$ (3.18)

###### Proof.

By a simple change of variables, we have, in more conventional notation,

$\int_{-\infty}^{\infty}\zeta\left(\sigma+i\,v\right)r^{\sigma+i\,v}dv=-i\int_{\sigma-i\,\infty}^{\sigma+i\,\infty}\zeta\left(s\right)r^{s}ds$
(3.19)

which we identify [7, Eq. II.4.(13b)] as a 2-sided Fourier transform of the
form

$h\left(t\right)=\frac{1}{2\,\pi\,i}\int_{\sigma-i\,\infty}^{\sigma+i\,\infty}\frac{{\mathrm{e}}^{s\,t}\,f\left(s\right)}{s}ds$
(3.20)

where $t=\ln(r)$, $h(\ln(r))=2\pi\,r\,\delta(r-n)$, and
$f(s)=2\pi\,s\,\zeta(s)$. The inverse transform of (3.20) is [7, Eq.
II.4.(13b)]

$f\left(s\right)=s\int_{-\infty}^{\infty}{\mathrm{e}}^{-s\,t}\,h\left(t\right)dt,\hskip
80.0pt\Re(s)>1,$ (3.21)

so first apply the change of variables $t=\ln(r)$ to find [6]

$f\left(s\right)=s\int_{0}^{\infty}r^{-1-s}\,h\left(\ln\left(r\right)\right)dr,$
(3.22)

and then identify the various components of (3.20) to obtain (with $n\neq 0$),

$\displaystyle\zeta\left(s\right)$
$\displaystyle=\int_{0}^{\infty}r^{-s}\,\delta\left(n-r\right)dr$ (3.23)
$\displaystyle=\sum_{n=1}^{\infty}1/n^{s},\hskip 80.0pt\Re(s)>1\,,$ (3.24)

the primary definition of $\zeta(s)$. ∎

######  Remark 3.1.3.

The identity (3.12) along with (3.18) presents an interesting example of a
non-constant function, all of whose derivatives vanish. See [9] for a
discussion about this point.

###  3.2 The case $\sigma<1$

As suggested in [1], we now treat any of the above entities as a contour
integral over a line $\Re(v)=\sigma$ in the complex $v$-plane where
$v=\sigma+it$ by rewriting any of the above in the more conventional form of
the contour integral representation (see (1.1))

$\int_{-\infty}^{\infty}\frac{\zeta\left(\sigma+i\,t\right)r^{\sigma+i\,t}}{\sigma+i\,t}dt=-i\int_{\sigma-i\,\infty}^{\sigma+i\,\infty}{\zeta\left(v\right)r^{v-1}}dv,$
(3.25)

and translate the contour (equivalent to invoking the Master Theorem) such
that $\sigma<1$ by accounting for the residues of the integrand so transited.
First, we encounter a pole at $v=1$ with residue $r$; secondly a pole at $v=0$
with residue $-1/2$ and consider various cases.

####  3.2.1 $r\neq n$

From the above, we obtain

$\displaystyle\int_{-\infty}^{\infty}\frac{\zeta\\!\left(\sigma+i\,t\right)r^{\sigma+i\,t}}{\sigma+i\,t}dt$
$\displaystyle=2\,\pi\left({\lfloor r\rfloor}-r\right),\hskip 20.0pt$
$\displaystyle 0<\sigma<1,$ (3.26) $\displaystyle=2\,\pi\left({\lfloor
r\rfloor}-r\right)+\pi,$ $\displaystyle\sigma<0,$ (3.27)

(3.26) being an exact analogue of (2.8) – there is no $\sigma$ dependence on
the right-hand side other than the specification of the range of
applicability, and it is invariant if $r:=r+m$. In exact analogy to subsection
3.1.1, differentiating first with respect to $\sigma$, then with respect to
$r$ in both of (3.26) and (3.27) produces

$\int_{-\infty}^{\infty}\zeta^{\left(1\right)}\\!\left(\sigma+i\,t\right)r^{\sigma+i\,t}dt=2\,\pi\,r\,\ln\\!\left(r\right),\hskip
20.0pt\sigma<1\,.$ (3.28)

Again, as in (2.16), by differentiating (3.26) or (3.27) with respect to $r$
we obtain

$\displaystyle\int_{-\infty}^{\infty}\zeta\left(\sigma+i\,t\right)r^{\sigma+i\,t}dt$
$\displaystyle=-2\,\pi r\hskip 20.0pt$ $\displaystyle\sigma<1\,,r\neq n,$
$\displaystyle=-\pi r$ $\displaystyle\sigma=1\,,r\neq n,$ (3.29)

in agreement with (2.18) by utilizing half the appropriate residue when
$\sigma=1$.

######  Remark 3.2.1.

Since $\zeta(\sigma+it)\approx-i/t+O(t^{0})$ if $\sigma=1$, the imaginary part
of the integrand contains the pole at $t=0$ and the imaginary integral that
contains this divergent term, vanishes by anti-symmetry, leaving the finite
result (3.29).

####  3.2.2 $r=n$

For the case $r=n$, from (3.14) plus the residues discussed above, we have

$\displaystyle\int_{-\infty}^{\infty}\frac{\zeta\\!\left(\sigma+i\,t\right)n^{\sigma+i\,t}}{\sigma+i\,t}\,dt$
$\displaystyle=\pi(2n-1)-2n\pi$ $\displaystyle=-\pi\hskip 20.0pt$
$\displaystyle 0<\sigma<1;$ (3.30) $\displaystyle=\,~{}0$
$\displaystyle\sigma<0,$ (3.31)

consistent with [1, Eqs. (4.3), (4.4) and (4.7)], all of which correspond to
the case $n=1$. Differentiating with respect to $\sigma$ gives

$\displaystyle\int_{-\infty}^{\infty}\frac{\zeta^{\left(1\right)}\\!\left(\sigma+i\,t\right)n^{\sigma+i\,t}}{\sigma+i\,t}dt-\int_{-\infty}^{\infty}\frac{\zeta\\!\left(\sigma+i\,t\right)n^{\sigma+i\,t}}{\left(\sigma+i\,t\right)^{2}}dt$
$\displaystyle=\pi\,\ln\\!\left(n\right),\hskip 20.0pt$ $\displaystyle
0<\sigma<1,$ (3.32) $\displaystyle=0$ $\displaystyle\sigma<0.$ (3.33)

As in (3.16), setting $n=1$ in (3.32) along with (3.33) gives the
generalization

$\int_{-\infty}^{\infty}\frac{\zeta^{\left(1\right)}\\!\left(\sigma+i\,t\right)}{\sigma+i\,t}dt=\int_{-\infty}^{\infty}\frac{\zeta\\!\left(\sigma+i\,t\right)}{\left(\sigma+i\,t\right)^{2}}\,dt,\hskip
20.0pt\forall\,\sigma.$ (3.34)

Again, following Section 2, the value of (3.29) when $r=n$ remains unclear,
since the definition of derivative at a point is indeterminate. These issues
are studied numerically in the following Section.

##  4 Numerical tests

Each of the integrals studied here is inherently oscillatory and therefore
difficult to evaluate numerically, particularly since some of them may
diverge. A simple way to explore the properties of such entities is to
transform each into a (infinite) summation by the elementary act of
subdividing the integration range into a large number of small, equal parts.
By studying the numerical convergence of the sum employing Cesàro summation
(known to provide a means of regularizing divergent – or difficult – sums), we
obtain a means of verifying, or at least increasing confidence, in the
identities that have been developed here (see Appendix A).

###  4.1 $\sigma>1$

(a) Cesàro approximation to the function defined in (4.1) with $\sigma=r=4$,
suggesting divergence of order $T/2$.

(b) Cesàro approximation to the function defined in (4.1) with
$\sigma=4,~{}r=3.9$, suggesting convergence to zero.

Figure 3: Numerical approximations to (4.1)

First, we consider the simple case $\sigma>>1$, (where the (oscillatory)
integral is expected to converge when $r\neq n$ and there are no issues
associated with analytic continuation), by setting $\sigma=4$ and comparing
the cases $r=4$ and $r=3.9$. According to (3.13), the integral

$\displaystyle{\underset{T\rightarrow\infty}{lim}}\int_{0}^{T}\Re\left(\zeta\\!\left(4+i\,t\right)r^{4+i\,t}\right)dt=0\,,\hskip
20.0ptr\neq n,$ (4.1)

but could be either zero or infinite if $r=n$, depending on how the derivative
is defined – see Section 2.4. Figure (3(a)) is suggestive that (4.1) diverges
to infinity of $\mathrm{O}(T^{1})$, at least when $r=4$. Similarly, when
$r=3.9$, Figure (3(b)) is consistent with (4.1) and lends credibility to that
identity, which states that the approximation should converge to zero.

(a) Cesàro approximation to the function defined in (4.2) with
$\sigma=1/2,~{}r=0.9,~{}r=1.1$ and $r=2.1$, suggesting convergence to $-\pi r$
as T increases.

(b) Cesàro approximation to the function defined in (4.2) with
$\sigma=1/2,~{}r=1$ and $r=2$, revealing a reasonable indication of
divergence. The two cases are numerically indistinguishable within graphical
resolution.

Figure 4: Numerical approximations to (4.2).

###  4.2 $0<\sigma<1$

It is also of interest to calculate the similar case where $\sigma<1$; that
is, from (3.29) we consider the identity

$\displaystyle\int_{0}^{\infty}\Re\left(\zeta\\!\left(\sigma+i\,t\right)r^{\sigma+i\,t}\right)dt=-\pi\,r\hskip
20.0ptr\neq n,$ (4.2)

for several different values of $r$ and $\sigma$. Before doing so, we rewrite
the finite version of (4.2) in several equivalent forms:

$\displaystyle\int_{0}^{T}\Re\left(\zeta\left(\sigma+i\,t\right)r^{\sigma+i\,t}\right)$
$\displaystyle
dt=\frac{r^{\sigma}}{2}\int_{-T}^{T}\zeta\left(\sigma-i\,t\,\right){\mathrm{e}}^{-i\,t\,\ln\left(r\right)}dt$
(4.3)
$\displaystyle=r^{\sigma}\,\int_{0}^{T}\cos\left(t\,\ln\left(r\right)\right)\zeta_{R}\left(\sigma+i\,t\right)-\sin\left(t\,\ln\left(r\right)\right)\zeta_{I}\left(\sigma+i\,t\right)dt$
(4.4)
$\displaystyle=\int_{0}^{T}{|\zeta\left(\sigma+i\,t\right)|^{2}}\,\cos\left(t\,\ln\left(r\right)+\arg\left(\zeta\left(\sigma+i\,t\right)\right)\right)\,dt,$
(4.5)

and consider the above as $T\rightarrow\infty$. Inspired by Figure 3(b), we
speculate that the integral itself, as a function of $T$, reflects the
periodic nature of the trigonometric terms of the integrand. That this is more
than speculation is illustrated in Figure 4(a), showing a slow, but steady
convergence to $-\,\pi r$ using several values of $r\neq n$ and $\sigma=1/2$.
In addition, the unexpected periodicity of the integral reappears, to be
studied in Section 4.4. In the case $r=n$, Figure 4(b) suggests that (4.2)
diverges of $\mathrm{O}(T^{1})$.

Figure 5: Cesàro approximations to the function defined in (4.2) with $r=2.1$
and various $0<\sigma\leq 1$, with $\sigma=1/3,~{}\sigma=3/4$ and $\sigma=7/8$
not shown because they are numerically indistinguishable, within the
resolution of the figure, from the case $\sigma=1/2$ selected. The arrows mark
integral multiples of a conjectured period $\rho$.

When a similar calculation is performed for different values of $0<\sigma\leq
1$ and constant $r$, the Cesàro estimate also converges to the expected limit
(4.2), which limit varies only if $\sigma=1$ as expected (see Remark 3.2.1 and
(3.29)). This is illustrated in Figure 5, which focusses on the Cesàro
estimates for several values of $\sigma$ and constant $r=2.1$. This figure
shows only the cases $\sigma=1/2$ and $\sigma=1$ because the others are
indistinguishable from the case $\sigma=1/2$ within graphical resolution and
again reveals a strong periodicity..

Figure 6: Study with $r$ close to $n$. Down pointing vertical arrows indicate
multiples of the solution points $t$ where $t\ln(1.01)=n\pi$, $n\geq 1$. Up
pointing arrow labels two cases when $r\approx n$.

Figure 7: This is a detailed comparison between the Cesàro estimates with
three values of $\sigma$ at constant $r=2.1$. It also shows that the partial
sum for $\sigma=-1/2$ crosses the asymptotic line in three places.

It is also of interest to question how (4.2) converges to its discontinuous
limit. Figure 6 shows that the convergence for $r=1.01$ and $r=1.001$ exists
but it takes longer to “turn over” as $r$ approaches unity. Significantly,
once it does “turn over”, it will approach the finite limit $-\pi r$, rather
than diverging. Since we shall shortly see that the oscillations in the
partial sum appear to be influenced by a periodicity $\rho=2\pi/\ln(r)$, when
$r=1$ the curve will never turn over, and (4.2) becomes infinite at $r=1$.
However, as Figure 6 shows, this does not lead to any insight concerning the
case $r\approx 2$, or indeed any other integer – see Figures 3(a) and 4(b).

In a final study of this case, Figure 7 presents a detailed view near a
minimum, of the Cesàro estimates with $\sigma=1/2$, $\sigma=0$ and
$\sigma=-1/2$, all of which, according to (4.2), should converge to the same
limit; as observed in this Figure they are effectively indistinguishable
within graphical resolution. We also note that the partial sum estimate for
$\sigma=-1/2$ is far more varied than is the same estimate for the integral
corresponding to $\sigma=1/2$ (shown below in Figure 11(a) of Section 4.4),
but it still intersects the asymptotic line $-2.1\pi$ near the minimum of the
Cesàro estimates.

Figure 8: A test of (3.28) with $\sigma=1/3,~{}r=3/2$.

Figure 9: This Figure shows the partial sum of both the difference of the two
integrals (red jagged curve) and the second integrand term itself (connected
by the arrow), as well as the Cesàro estimate (dashed curve), all of which are
tending to zero.

###  4.3 Derivatives

To test the identities involving derivatives, consider Figure 8, a test of
(3.28) showing convergence to the expected limit. Mindful of the ambiguity
discussed with respect to (3.16), we next consider Figure 9 showing that the
difference of the two integrals (when $\sigma>1$) is tending to zero as
expected. Figure 10 shows that (3.10) and (3.15) also appear to be approaching
their respective limits as predicted.

(a) Test of (3.10) (i.e., $\sigma>1$ and $r\neq n$), calculated as a
difference of two integrals. (b) Test of (3.15) (i.e., $\sigma>1$ and $r=n$),
calculated as a difference of two integrals.

Figure 10: This Figure tests (3.10) and (3.15), both valid for $\sigma>1$.

###  4.4 Periodicity

The periodicity of the estimates (e.g. Figure 5), suggests the existence of a
relationship with the periodicity of the integrand trigonometric functions,
with a period being integral multiples of a quantity $\rho$ that satisfies the
identity $2m\pi=\rho\ln(2.1)$, that is, if $m=1$, $\rho=8.47$ – see (4.4). The
arrows in Figure 5 indicate an approximate observed periodicity of the Cesàro
estimate equal to $15\rho$, with a small phase shift as $T$ increases. At a
higher multiple of the initial period ($\sim 180\rho$) the minimum appears to
be shifted slightly. This is examined in more detail in Figure 11.

(a) The small down arrow marks the point corresponding to the fifteenth
harmonic of the assumed period $\rho$. The two other arrows indicate the
minimum of the respective Cesàro approximation when $\sigma=1/2$ and
$\sigma=1$. The partial sum corresponding to the numerical integration is also
shown crossing the line ( $-2.1\pi$ ) that marks the asymptotic limit.

(b) The small down arrows mark the points at a multiple of the harmonic in
Figure 11(a). The two other arrows indicate the minimum of the respective
Cesàro approximation when $\sigma=1/2$ and $\sigma=1$. The partial sum
corresponding to the numerical integration up to the point $T$, is also shown
crossing the lines ($-1.05\pi$) and ($-2.1\pi$ ) that mark the asymptotic
limit for $\sigma=1$ and $\sigma=1/2$ respectively. When $\sigma=1/2$, this
occurs at $T=1549.31$.

Figure 11: Focus on the Cesàro approximations to the function defined in (4.2)
with $r=2.1$ and various $0<\sigma\leq 1$. The tested cases
$\sigma=1/3,~{}3/4$ and $\sigma=1/2$ are indistinguishable within graphical
resolution.

Because the minimum of the Cesàro estimate also appears to numerically
coincide with the asymptotic ($T\rightarrow\infty$) value of the integral
($-2.1\pi=-6.597$), Figure 11(a) isolates the region $T\approx 15\rho$ where
it is shown that the near coincidence with the asymptotic result $-2.1\pi$ is
close to, but not exactly in phase with the suspected periodicity within the
accuracy of the numerical estimates used. Encouraged to speculate further by
these near coincidences, consider Figure 11(b) where the same estimates are
presented when $T\approx 183\rho$. Here we find a very near-coincidence
between the points where the partial sum crosses the asymptotic line
($-2.1\pi$) and the location of the “harmonic” $183\rho$. For comparison, the
location of the $182^{nd}$ harmonic is indicated by its own arrow and we note
the the minimum of the Cesàro estimate is bounded by those two points.

Although the Cesàro estimate appears to coincide with the asymptotic line in
all these Figures, in fact careful study shows that it always lies a small
distance above that line and by reasonable interpolation, it is possible to
determine the point and distance of closest approach. These points are marked
by large down arrows in Figure 11 and presented quantitatively in Figure 12
where we can see that the distance of closest approach decreases as $T$
increases, as expected of a converging approximation. Furthermore, by
reasonable interpolation of the partial sums as they cross the asymptotic
line, it is possible to test the possibility that the integral itself is
periodic in integer multiples of the variable $\rho$; although the period
between successive intersections was found to be within a few percent of the
integral ratio $1:2:3\cdots$, the values were never close enough to perfect
integers to validate such a hypothesis, and, after examining several
variations with different values of $r$, it seems that any postulated
universal multiplier increases with $r$ and does not appear to exist.

Figure 12: For each minimum marked in Figure 5 this shows the distance between
the Cesàro estimate and the asymptotic line $-2.1\pi$. See down arrows in
Figure 11.

##  5 Correlations

As has been seen, the finite version of the integral $Z^{\prime}(\sigma,r)$ is
oscillatory with a periodicity that is approximately constant if $r\neq n$,
suggesting that segments of the integral itself are periodic and therefore
correlated. Since the integral reflects the oscillatory nature of the
integrand, it is worthwhile to conjecture that the integrand itself possesses
corresponding segments with periodically correlated values. This could be a
reflection of either (or both) of the cosine or absolute value factors
appearing in (4.5).

Specifically, if, as observed (see e.g., Figure 17(b)), a partial sum and
therefore its corresponding proper integral itself, is periodic, that is, in
general, if

$\int_{L_{1}}^{L_{2}}f\left(t\right)dt\approx\int_{L_{1}+\rho}^{L_{2}+\rho}f\left(t\right)dt,\hskip
50.0pt\forall L_{1},L_{2},$ (5.1)

then

$\int_{L_{1}}^{L_{2}}f\left(t+\rho\right)-f\left(t\right)dt\approx 0\,,$ (5.2)

and one possibility is that $f(t)$ and $f(t+\rho)$ are correlated to some
extent. With reference to (4.5), one expects a considerable degree of
periodicity attached to the cosine function; however, more intriguing is the
possibility that a correlation exists between elements of $\zeta(\sigma+it)$
lying on corresponding offset (i.e., shifted) segments of the imaginary line
$\sigma+it$.

(a) This shows overlain segments of the argument of (4.5), each shifted
relative to $t=0$ by the amounts $\rho$ indicated. In all cases, $r=2.1$.

(b) This shows overlain segments of the function
$\left|\zeta(1/2+it)\right|^{2}$, each shifted relative to $t=0$ by the
amounts $\rho$ indicated.

Figure 13: This Figure overlays two quantities, each shifted by $t\rightarrow
t+\rho$ shown.

Again, inspired by Figures 5, 7 and 17(b), Figure 13, which overlays three
segments of the imaginary line, each of length 126 (see Figure 11) and each
offset from $t=0$ by the quantity $\rho$ indicated, suggests that a
correlation may, in fact, exist. This can be quantitatively explored by
devising a coefficient to measure any correlation that may exist between
segments of the $Z^{\prime}(\sigma,r)$ integrand as a function of the offset
quantity $\rho$.

For any two continuous functions $f(t)$ and $g(t)$, we borrow from statistics,
and define a correlation coefficient $\mathrm{Cor}(f,g)$ by

$\mathrm{Cor}(f,g)=\frac{\mathrm{Cov}(f,g)}{\sqrt{\mathrm{Var}(f)\,\mathrm{Var}(g)}}$
(5.3)

where the “Covariance” between two functions $f(t)$ and $g(t)$ is defined by

$\mathrm{Cov}(f,g)\equiv E(fg)-E(f)E(g),$ (5.4)

the “Variance” by

$\mathrm{Var}(f)\equiv E(f^{2})-E(f)^{2}$ (5.5)

in terms of the fundamental quantity “Expectation” defined for any function
$h(t)$ by

$E(h)\equiv\frac{1}{L_{2}-L_{1}}\int_{L_{1}}^{L_{2}}h(t)\,dt\,.$ (5.6)

Effectively, the correlation coefficient measures the normalized (scaled
between $\pm 1$) correlation between the values of $f(t)$ and $g(t)$ averaged
over the interval $\left[L_{1},L_{2}\right]$, with the statistical average
operator replaced here by a numerical integration operator. If
$\mathrm{Cov}(f,g)$ is close to $\pm 1$, the functions $f(t)$ and $g(t)$ are
strongly correlated/anticorrelated; if $\mathrm{Cov}(f,g)$ is close to zero,
they are uncorrelated; otherwise the strength of the (anti)correlation becomes
a subject of statistical study and opinion. Since we are not dealing here with
statistical correlations between individual Cesàro summation elements, it is
worth noting an adage that encapsulates varied opinion on this subject:
correlation coefficients between 0.5 and 0.7 indicate variables that are
moderately correlated. See also [10].

Figure 14: This Figure shows the correlation coefficient between
$f(t)=\left|\zeta(1/2+it)\right|$ and $f(t)=\left|\zeta(1/2+i(t+\rho))\right|$
as a function of $\rho$, between offset intervals starting at $t=L_{1}$ as
shown. Values above 0.5 are considered to be moderately correlated.

Figure 15: This Figure shows the correlation coefficient between
$f(t)=\left|\zeta(3/4+it)\right|$ and $f(t)=\left|\zeta(3/4+i(t+\rho))\right|$
as a function of $\rho$.

In the light of the observations discussed previously, and based on Figure 7,
in the following, set $L_{2}=L_{1}+126$. Figure 15 shows the correlation
coefficient $\mathrm{Cor}(f,g)$ between $f(t)=\left|\zeta(1/2+it)\right|$ and
$g(t)=\left|\zeta(1/2+i(t+\rho))\right|$ as a function of $\rho$, evaluated
over two contiguous pairs of segments, one starting at $L_{1}=0$, the other
starting at $L_{1}=126$. This indicates that over a very narrow range of
offset – notably $\rho=126.1\pm 0.6$ – there exist offset segments, between
which the average value of $\zeta(1/2+it)$ compared between each segment, is
moderately correlated. The fact that such contiguous segments exist and
moderate correlation can be found for very precise values of the offset
$\rho$, suggests that this is not a random statistical artifact. Therefore, it
is reasonable to theorize that there exist other segments of the line $1/2+it$
among which the average value of the function $\left|\zeta(1/2+it)\right|$ is
moderately auto-correlated [11]. This observation may in turn reflect on the
distribution of zeros of the function $\zeta(1/2+it)$ (see [12]).

Since correlations between $\zeta(\sigma+it)$ and offset $\sigma$ have been
reported elsewhere [13, Section 6.4.1], all of the above suggests that we
evaluate the correlation coefficient for a wider range of $\sigma$ over a
wider range of $\rho$. This is performed in Figure 15 showing that moderate
correlation exists over a much broader range of $\rho$ than the previous
exercise would lead us to believe. In this Figure, we see that moderate
correlation exists for offsets of $\rho=126.1\pm 0.6$ and $\rho=136.5\pm 0.7$
all of which suggests that there exists an underlying offset of $\rho\sim
10.1$ between correlated average values of $\zeta(\sigma+it)$. Qualitatively,
the separation of peaks in Figure 16 lends plausibility to this conjecture.
The study of offset (i.e., shifted) moments of the zeta function involving
integrals similar to those studied here also can be found in the literature
(e.g. [14], [15]).

Figure 16: A comparison of $\left|\zeta(3/4+i(t+\rho))\right|$ with two
different offsets over a small range of $t$. The corresponding comparisons for
the real and imaginary parts of $\zeta(\sigma+it)$ are given in Appendix C,
Figures 18 and 19

.

##  6 Summary

In this work, the functions $Z(\sigma,r)$ and $Z^{\prime}(\sigma,r)$ have been
studied, both by analysis and numerical experimentation. As discovered both
here and elsewhere, these functions possess the interesting property that for
certain values of the parameter $r$, the value of the function depends
continuously on its limiting value as $r$ approaches an integer $n$, but
discontinuously at the point of discontinuity $r=n$. In particular, the
traditional limiting value of either function as $r\rightarrow n$ is different
from its value at $r=n$, demonstrating that both functions are discontinuous
and the traditional definition $\underset{r\rightarrow n}{\lim}$ is invalid,
but only at the limit point. With respect to the function $Z(\sigma,r)$, the
value at the point of discontinuity $r=n$ lies naturally, rather than by
decree, halfway between its value(s) as the limit point is approached from
either direction. This property was verified numerically.

The derivative function $Z^{\prime}(\sigma,r)$ was also shown to have a
similar property, notably that, for a certain range of the parameter $\sigma$,
it vanishes everywhere except at points $r=n$ where it is divergent. In this
respect it represents one tine of the Dirac comb function, without the
necessity of invoking limits or test functions. Again, this was verified
numerically – the numbers speak for themselves.

A very significant outcome of this study was the demonstration of the power of
a modified Cesàro summation to approximate an improper integral and to verify
analytic predictions. It is well-known (e.g., see Figures 16 and 17(a)) that
the Riemann zeta function fluctuates wildly, but somehow the Cesàro summation
tames the fluctuations and exposes properties that would otherwise have
remained well-hidden. Certainly, without the Cesàro approximation, the quasi-
periodic nature of $Z(\sigma,r)$ and the offset property of
$\left|\zeta(1/2+it)\right|)$ would be unrecognized.

Having brought all these interesting properties to light, it is necessary to
acknowledge that many questions remain, all of which are outside the scope of
this study. Chief among them leads one to question “Why is Cesàro
approximation so effective?” Along the same lines, we observe that the Cesàro
approximation appears to always approach its asymptotic limiting value from
one direction – is this significant and is there a reason? With respect to the
fact that the modified Cesàro summation was originally introduced as a means
of approximating an improper integral, it is necessary to recognize that
although $T\sim 4000$, the maximum used here, takes us a reasonable distance
along the number line, it is still a long way to infinity. Thus there is no
assurance of the validity of the underlying assumption – the numerical
observations presented will continue to behave as they have done here, as
$T\rightarrow\infty$.

When studying the periodic nature of the functions, it was first observed that
the offset parameter $\rho=126$ yielded a reasonable suggestion of the
existence of an underlying correlation, but that this choice is probably
serendipitous and reflects a deeper periodicity of $\sim 10$. If that is the
case, the connection between a very clear Cesàro periodicity signal and a
moderate underlying auto-correlation of the integrand function remains to be
investigated. The underlying reason for the numerical value of any of the
observed periodicities is not evident, and so it must be finally noted that
the numerical experiments reported here merely scratch the surface of further
study needed to verify these observations.

##  7 Caveats

The calculations and analysis in this work are based on two fundamental
premises:

  * •

Glasser’s Master Theorem (or, equivalently the translation of contour
integrals) is valid only if the integrand functions vanish appropriately at
the integration end-points;

  * •

The numerical integrations were performed initially using [5, Maple] and later
[6, Mathematica] when it was found that the latter executed orders of
magnitude faster than the former and gave the same answers. However, the
overall validity is dependent on the accuracy of algorithms built into these
two computer codes on which our results are dependent.

##  8 Acknowledgement

Professor Christophe Vignat (Tulane University) commented perceptively on the
previous work [1] and that comment led directly to the present study. He is
also the author of the challenge Appendix B; we are grateful for his
contribution.

##  9 Funding

All expenses associated with this work were funded by the authors.

## References

  * [1] Michael Milgram.  An extension of Glasser’s master theorem and a collection of improper integrals many of which involve Riemann’s Zeta function.  J. Classical Analysis (to be published), 2024.  https://arxiv.org/abs/2402.17586.
  * [2] M. L. Glasser.  A remarkable definite integral, 2013.  https://arxiv.org/abs/1308.6361v2.
  * [3] Michael Milgram.  Determining the indeterminate: On the evaluation of integrals that connect Riemann’s, Hurwitz’ and Dirichlet’s Zeta, Eta and Beta functions.  2021\.  available from https://arxiv.org/abs/2107.12559.
  * [4] H.M. Edwards.  Riemann’s Zeta Function.  Dover, 2001.
  * [5] Maplesoft, a division of Waterloo Maple Inc., version 2023.  Maple.
  * [6] Wolfram Research, Champaign, Illinois.  Mathematica, version 13.2, 2023.
  * [7] B. Van Der Pol and H. Bremmer.  Operational Calculus based on the two-sided Laplace Integral.  Cambridge University Press, 1959.
  * [8] I.S. Gradshteyn and I.M. Ryzhik.  Tables of Integrals, Series and Products, corrected and enlarged Edition.  Academic Press, 1980.
  * [9] https://math.stackexchange.com/questions/240026/all-derivatives-zero-at-a-point-implies-constant-function.
  * [10] https://stats.stackexchange.com/questions/233606/exact-meaning-of-correlation-coefficient.
  * [11] Michael J. Curran.  Correlations of the Riemann zeta function, 2023.  https://doi.org/10.48550/arXiv.2303.10123.
  * [12] Siegfred Alan C. Baluyot, Daniel Alan Goldston, Ade Irma Suriajaya, and Caroline L. Turnage-Butterbaugh.  An unconditional Montgomery theorem for pair correlation of zeros of the Riemann zeta function, 2023.  https://doi.org/10.48550/arXiv.2306.04799.
  * [13] Michael Milgram.  An integral equation for Riemann’s Zeta function and its approximate solution.  Abstract and Applied Analysis, /2020/1832982(1832982), May 2020\.  https://doi.org/10.1155/2020/1832982.
  * [14] Vorrapan Chandee.  On the correlation of shifted values of the Riemann zeta function, 2009\.  https://doi.org/10.4153/S0008414X23000548.
  * [15] Nathan Ng, Quanli Shen, and Peng-Jie Wong.  Shifted moments of the Riemann zeta function, 2022.  https://arxiv.org/abs/2206.03350.
  * [16] Cesàro Summation, March 2024.  https://en.wikipedia.org/wiki/Cesaro_summation.
  * [17] I.I. Volkov.  Cesàro_summation_methods&oldid=26199.  Encyclopedia of Mathematics, 2020.  http://encyclopediaofmath.org/index.php?title=Ces%C3%A0ro_summation_methods&oldid=26199.
  * [18] E.C. Titchmarsh and D.R Heath-Brown.  The Theory of the Riemann Zeta-Function.  Oxford Science Publications, Oxford, Second edition, 1986.

##  Appendix A Appendix: Cesàro Summation - a primer

Cesàro summation [16] assigns a value to an improper integral or sum that is
not necessarily convergent in the usual sense. In the first case, the Cesàro
summability of an integral involves the limit of the means of its partial
integrals; in the second case it involves the limit of the means of its
partial sums. Here, we study an improper integral in a different way, by first
converting it into a sum, utilizing simple subdivision and numerical
integration, and then employ Cesàro summation to evaluate the sum. In
particular we employ the Cesàro algorithm $(C,1)$ [17], defined as the limit,
as $J$ tends to infinity, of the sequence of arithmetic means of the first $J$
partial sums of the series. Since we are here interested in improper
integrals, some of which possess debatable convergence status, by transforming
each integral into a corresponding sum and observing a large number of partial
sums, we obtain, if convergent, a valid estimate of its value, and, if
divergent, an accepted regularization, which may, or may not, tend to
infinity. To be specific, consider an integral

$H(T)=\int_{0}^{T}h(t)dt$ (A.1)

where we are interested in the limit $T\rightarrow\infty$ . Subdivide the
integral into small parts defined by

$h\left(t_{j}\right)=\int_{t_{j}}^{t_{j}+\delta}h\left(t\right)dt$ (A.2)

so that

$H\left(t_{m}\right)=\mathop{\kern 0.0pt\mathop{\kern
0.0pt\sum}\limits_{j=1}}\limits^{m}h\left(t_{j}\right)$ (A.3)

where $t_{1}=0$, $t_{J}+\delta=T$ and $H(T)=H(t_{J})$. As employed here, we
set $\delta=1$ and evaluate each of the elements $h(t_{j})$ by numerical
integration over the appropriate interval to arrive at an accurate numerical
estimate of the value of the integral $H(t_{m})$ over some interval defined by
the choice of $j\leq m$. We now form partial sums defined by

$P_{k}=\mathop{\kern 0.0pt\mathop{\kern
0.0pt\sum}\limits_{m=1}}\limits^{k}H\left(t_{m}\right),\hskip 50.0ptk\leq J,$
(A.4)

and form the running average of each partial sum

$H_{n}=\frac{1}{n}\mathop{\kern 0.0pt\mathop{\kern
0.0pt\sum}\limits_{k=1}}\limits^{n}P_{k}\,.\hskip 50.0ptn\leq J$ (A.5)

(a) Individual integration elements $h(t_{j})$.

(b) The corresponding partial sums $P_{k}$. These also equate to the numerical
value of the integral (4.3) as a function of its upper limit $T$. The
asymptotic limit $-2.1\pi$ is also shown.

Figure 17: A comparison of individual elements and partial sums. The average
of the partial sums $P_{k}$ can be found in Figure 4(a)

.

If the sum is convergent, when $n=J$, $H_{J}$, the Cesàro sum, is an accurate
numerical estimate [7, VI.8] of the sum of the series (A.3) and hence of the
finite integral (A.1) and our interest focusses on $H_{J}$ as
$J\rightarrow\infty$. The advantage of evaluating the integral in this manner
is that it affords the analyst a simple means to study the properties of the
integral as it converges to its upper limit, and, in the case of violently
oscillating integrands, averaging the running partial sums tends to smooth any
associated noise. If the integral diverges, it is still possible that the sum
$H_{n}$ will converge to some finite value as $n\rightarrow J$ as $J$ itself
increases, thereby yielding one possible regularization of a divergent series
and its underlying divergent integral – the sum of the series being defined
(i.e., regularized) by the average of its partial sums. It is educational to
consider an example examined in this study.

Figure 17(a) demonstrates that the individual elements $h(t_{j})$ of the
subdivision of the integral (4.2) under consideration vary wildly as expected
for the choice $\sigma=1/2$ and $r=2.1$. Figure 17(b) shows that the partial
sum of these elements varies considerably less violently than do the
individual elements themselves. Since the partial sum equates to the numerical
evaluation of the integral with varying upper limit, in this case we see that
the finite integral periodically intercepts its own asymptotic value and is
itself periodic. Finally, as presented previously in Figure 4(a), we see that
the act of averaging the partial sums has completely washed out the noise,
suggesting that the integral is dominated by the periodic components of its
integrand. This is studied in Section 4.

######  Remark A.0.1.

We note that it is the rare analyst who would pause to consider the average
characteristics of the individual elements of a numerical integration and it
is only through the use of Cesàro summation that one might consider such an
examination.

##  Appendix B Appendix: An enigmatic derivation

As an alternative to the derivation presented in Section 2.2, here we consider
an attempted independent derivation of (2.6) and (2.7) by direct integration
of an identity equivalent to the well-known [18, Eq. (2.1.4)], and convergent,
integral representation

$\displaystyle\frac{\zeta\\!\left(\frac{3}{2}-i\,t\right)}{\frac{3}{2}-i\,t}$
$\displaystyle=\int_{1}^{\infty}\frac{{\lfloor
x\rfloor}-x+\frac{1}{2}}{x^{\frac{5}{2}-i\,t}}dx+\frac{1}{\left(\frac{3}{2}-i\,t\right)\left(\frac{1}{2}-i\,t\right)}+\frac{1}{3-2\,i\,t}$
(B.1) $\displaystyle=\int_{1}^{\infty}\frac{{\lfloor
x\rfloor}-x+\frac{1}{2}}{x^{\frac{5}{2}-i\,t}}dx+\frac{\frac{1}{2}+i\,t}{t^{2}+{1}/{4}}-\frac{\frac{3}{2}+i\,t}{2\left(t^{2}+{9}/{4}\right)}\,.$
(B.2)

First, pre-multiply by $r^{-i\,t}$ and integrate (B.2), leading us to consider
the integral

$\displaystyle\int_{-\infty}^{\infty}\frac{\zeta\\!\left(\frac{3}{2}-i\,t\right)r^{-i\,t}}{\frac{3}{2}-i\,t}dt$
$\displaystyle=\int_{1}^{\infty}\frac{\left({\lfloor
x\rfloor}-x+\frac{1}{2}\right)}{x^{\frac{5}{2}}}\int_{-\infty}^{\infty}\left(\frac{x}{r}\right)^{i\,t}dt\,dx$
$\displaystyle+\int_{-\infty}^{\infty}\frac{r^{-i\,t}\left(1/2+i\,t\right)}{t^{2}+1/4}dt-\frac{1}{2}\int_{-\infty}^{\infty}\frac{r^{-i\,t}\left(3/2+i\,t\right)}{t^{2}+9/4}dt$
(B.3)

where the double integral operators in the first (right-hand side) term have
been transposed. In a fashion similar to (1.8) and (1.9), with respect to the
latter two integrals, we find

$\displaystyle\int_{-\infty}^{\infty}\frac{r^{-i\,t}\left(\frac{1}{2}+i\,t\right)}{t^{2}+\frac{1}{4}}dt$
$\displaystyle=\frac{1}{2}\int_{-\infty}^{\infty}\frac{\cos\left(t\,\ln\left(r\right)\right)+2\,t\,\sin\left(t\,\ln\left(r\right)\right)}{t^{2}+{1}/{4}}dt$
$\displaystyle=\frac{2\,\pi}{\sqrt{r}}$ $\displaystyle\mathrm{if}~{}r>1;$
$\displaystyle=2\,\pi$ $\displaystyle\mathrm{if}~{}r=1;$ $\displaystyle=0$
$\displaystyle\mathrm{if}~{}0<r<1\,$ (B.4)

and

$\displaystyle\frac{1}{2}\int_{-\infty}^{\infty}\frac{r^{-i\,t}\left(\frac{3}{2}+i\,t\right)}{t^{2}+\frac{9}{4}}dt$
$\displaystyle=\frac{1}{2}\int_{-\infty}^{\infty}\frac{\frac{3}{2}\,\cos\left(t\,\ln\left(r\right)\right)+t\,\sin\left(t\,\ln\left(r\right)\right)}{t^{2}+{9}/{4}}dt$
$\displaystyle=\frac{\pi}{r^{\frac{3}{2}}}$ $\displaystyle\mathrm{if}~{}r>1;$
$\displaystyle=\pi$ $\displaystyle\mathrm{if}~{}r=1;$ $\displaystyle=0$
$\displaystyle\mathrm{if}~{}0<r<1\,.$ (B.5)

We now consider the first (right-hand side) term in (B.3) and, making use of
the identity (Maple, Mathematica)

$\int_{-\infty}^{\infty}\cos\\!\left(t\,\ln\\!\left(\frac{x}{r}\right)\right)dt=2\,\pi\,\delta\\!\left(\ln\\!\left(\frac{x}{r}\right)\right)$
(B.6)

along with the change of variables $\ln(\frac{x}{r}):=x$, we obtain

$\int_{1}^{\infty}{x^{-\frac{5}{2}}}\left({\lfloor x\rfloor}-x+\frac{1}{2}\right)\,\int_{-\infty}^{\infty}\left(\frac{x}{r}\right)^{i\,t}dtdx$ |
---|---
$=\int_{1}^{\infty}{x^{-\frac{5}{2}}}\left({\lfloor x\rfloor}-x+\frac{1}{2}\right)\,\int_{-\infty}^{\infty}\cos\\!\left(t\,\ln\\!\left(\frac{x}{r}\right)\right)dtdx$ |
$=\,-{\pi}{r^{-\frac{3}{2}}}\int_{-\ln\,\left({r}\right)}^{\infty}\left(2\,r\,{\mathrm{e}}^{x}-2\,{\lfloor r\,{\mathrm{e}}^{x}\rfloor}-1\right){\mathrm{e}}^{-3x/2}\,\delta\\!\left(x\right)dx$ |
$=\,-{\pi}{r^{-\frac{3}{2}}}\left(2\,r-2\,{\lfloor r\rfloor}-1\right)$ | $\mathrm{if}~{}r>1$
$={\pi}{n^{-\frac{3}{2}}}$ | $\mathrm{if}~{}r=n,~{}n\geq 1;$
$=0$ | $\mathrm{if}~{}0<r<1\,.$
(B.7)

######  Remark B.0.1.

Nowhere in this derivation does $r=n$ suggest the existence of a special
discontinuous case, and, in all cases, particularly (B.7), the case $r=n$
simply corresponds to the reduction $r\rightarrow n$ of the more general case
corresponding to $r>1$. In contrast, see Remark 2.2.1.

Putting it all together, this derivation yields the prospective, but aberrant,
identity

$\displaystyle\int_{-\infty}^{\infty}\frac{\zeta\\!\left(\frac{3}{2}-i\,t\right)}{\frac{3}{2}-i\,t}\,$
$\displaystyle r^{\frac{3}{2}-i\,t}dt\hskip 60.0pt$
$\displaystyle=2\,\pi\,{\lfloor r\rfloor}$ $\displaystyle\mathrm{if}~{}r>1\,;$
$\displaystyle=2\,\pi\,n$ $\displaystyle\mathrm{if}~{}r=n,n\geq 1;$
$\displaystyle=0$ $\displaystyle\mathrm{if}~{}0<r<1\,.$ (B.8)

######  Remark B.0.2.

Note that

  * •

(i) the derivations (2.7) and (B.8) disagree when $r=n$;

  * •

(ii) the disagreement extends to the numerical evaluation presented in Figure
1, and

  * •

(iii) this derivation employs the traditional definition of the floor function
(Remark 1.2.1).

##  Appendix C Appendix: Two Figures

The following two figures present the real and imaginary components of the
functions shown in Figure 16.

Figure 18: A comparison of $\Re\zeta(3/4+i(t+\rho))$ with two different
offsets over a small range of $t$. Figure 19: A comparison of
$\Im\zeta(3/4+i(t+\rho))$ with two different offsets over a small range of
$t$.


longtable Ginwidth=,height=,keepaspectratio

# Editorial: Guidelines and Best Practices to Share Deidentified Data and Code

Nicholas Jon Horton
Department of Mathematics and Statistics, Amherst College
and
Sara Stoudt
Department of Mathematics, Bucknell University

(May 28, 2024)

###### Abstract

No abstract for editorial

Keywords: no keywords for editorial

##  1 Introduction

In 2022, the _Journal of Statistics and Data Science Education_ (JSDSE)
instituted augmented requirements for authors to post deidentified data and
code underlying their papers (Horton et al. 2022). These changes were prompted
by an increased focus on reproducibility and open science (National Academies
of Sciences, Engineering, and Medicine 2019). A recent review of data
availability practices noted that “such policies help increase the
reproducibility of the published literature, as well as make a larger body of
data available for reuse and re-analysis” (Federer et al. 2018).

_JSDSE_ values accessibility as it endeavors to share knowledge that can
improve educational approaches to teaching statistics and data science.
Because institution, environment, and students differ across readers of the
journal, it is especially important to facilitate the transfer of a journal
article’s findings to new contexts. This process may require digging into more
of the details, including the deidentified data and code used to support the
article’s findings.

Two years on, we felt that it was valuable to provide our readers and authors
with a review of why the requirements for code and data sharing were
instituted, summarize ongoing trends and developments in open science, discuss
options for data and code sharing, and share advice for authors.

##  2 Why is data and code sharing necessary?

Why the push for the sharing of data and code? Data sharing, as well as the
sharing of the code for the processing and analysis steps, is a necessity if
readers or other researchers are to reproduce findings published in the
journal. Beyond building trust in a particular article’s findings, the sharing
of deidentified data can help others in their own studies.

An editorial in _PLOS ONE_ (2019) states that:

“Data availability allows and facilitates:

  1. 1.

Validation, replication, reanalysis, new analysis, reinterpretation or
inclusion into meta-analyses;

  2. 2.

Reproducibility of research;

  3. 3.

Efforts to ensure data are archived, increasing the value of the investment
made in funding scientific research;

  4. 4.

Reduction of the burden on authors in preserving and finding old data, and
managing data access requests;

  5. 5.

Citation and linking of research data and their associated articles, enhancing
visibility and ensuring recognition for authors, data producers and curators.”

Funding bodies have been requiring researchers to share their data as well. In
August 2022, the White House Office of Science and Technology Policy directed
federal agencies to update their data-sharing policies such that all
federally-funded research data would have to be shared freely by 2025 (DuBois
et al. 2023). The National Institute of Health’s Generalist Repository
Ecosystem Initiative has taken a comprehensive approach to foster data and
workflow sharing (National Institutes of Health Office of Data Science
Strategy 2023).

But it’s not just deidentified data that are needed. As but one example of the
importance of code sharing, Oza (2023) found that when a large group of
biologists analyzed the same dataset, their results were widely divergent.

A NASEM report on _Reproducibility and Replicability in Science_ concluded
that “Journal editors should consider ways to ensure reproducibility for
publications that make claims based on computations, to the extent ethically
and legally possible (page 2)” (National Academies of Sciences, Engineering,
and Medicine 2019).

A recent editorial in the Journal of the American Statistical Association
(JASA), the flagship journal of the American Statistical Association (ASA)
noted that their reproducibility process “fosters a culture of transparency
and accountability that is critical to nurture the trust placed by the lay
public in scientific research” (Wrobel et al. 2024).

We anticipate that other entities will be moving in this direction in the
future.

##  3 How have other journals responded?

In the past ten years, many top journals have added requirements to support
reproducible research, including policies on data sharing. In March 2014,
_PLOS ONE_ started requiring that the supporting data for the results in its
publications be shared (Federer et al. 2018). The _Harvard Data Science
Review_ released a data sharing requirement in 2021 (Harvard Data Science
Review 2021).

In September 2016, _Nature_ started to require that data availability
statements be included as part of its publications. Although this policy
doesn’t require that the data actually be made available, the intention of the
data availability statement is to make the authors’ decision to share or not
share the data more transparent.

The ASA released recommendations in 2018 for their journals that are jointly
published with Taylor & Francis (this set of journals includes _JSDSE_). These
recommendations encourage data sharing and the inclusion of data availability
statements (Wasserstein 2018).

In 2016, _JASA_ began to require both code and data at the revision stage of
submissions to the “Applications and Case Studies” section (JASA
Reproducibility Initiative 2019). _JASA_ editors extended this requirement to
the revision stage of submissions to the “Theory and Methods” section in 2021.

##  4 How best to share deidentified data

###  4.1 What datasets can be shared?

Many journals’ data sharing requirements reference a “minimum” or “minimal”
dataset that is required to be shared. For example, _PLOS ONE_ considers a
minimal dataset as one that has “the data required to replicate all study
findings reported in the article, as well as related metadata and methods”
(PLOS Journals 2019) while _Nature_ ’s minimum dataset is one that is
“necessary to interpret, verify and extend the research in the article,
transparent to readers” (Nature Portfolio 2019). The FAIR Data Principles of
data sharing request that these shared datasets are Findable, Accessible,
Interoperable , and Reusable with the goal of increasing “knowledge discovery
and innovation” (Wilkinson et al. 2016).

At times, investigators have gone back to their Institutional Review Board
(IRB) to seek approval for requests to retroactively share deidentified data.
At the heart of such a request is the creation of a “minimal” dataset without
any direct or indirect identifiers.

As one example, this process was undertaken to make the data from the Health
Evaluation and Linkage to Primary Care (HELP) study (Samet et al. 2003)
available on the Center for Open Science’s Open Science Framework (OSF)
(Horton 2024).

However, sharing a “minimum” dataset that just elides identifiers may not be
sufficient to avoid reidentification. When creating a “minimum” dataset it may
be necessary to only share a proper subset of the attributes recorded on
individual subjects to minimize risk of de-anonymization.

Moving forward, researchers need to be thinking about data sharing when they
begin to design their studies. The DMP Tool offers useful guidance for
investigators designing data management and sharing plans (California Digital
Library 2024).

Other approaches may be needed, depending on the study. Consider a
hypothetical qualitative research example where students are being interviewed
before graduating to find out about their trajectory through a data science
program. They are asked a series of questions, and their responses are
transcribed. The full text of the interview would reveal a lot of personally
identifiable information. However, the transcripts are often coded by
researchers as a pre-processing step before analysis. These codes are what are
used in the analysis phase of the project, so having access to the coded data
rather than the raw transcript data would be enough to reproduce the work.
_JSDSE_ has published papers involving the analysis of qualitative data
(Lesser & Santos 2024) where the coded dataset is shared (and not the full
transcripts, video recordings, etc.). DuBois et al. (2023) considers a similar
question.

Many journals’ data sharing requirements provide caveats and exceptions for
situations where data sharing may be problematic, especially with respect to
privacy, and leave room for more controlled access. Similar policies operate
for _JSDSE_ (see Section 6.3).

###  4.2 What file formats and licensing?

Making data “interoperable,” as the FAIR Data Principles advocate for,
includes sharing it in a non-proprietary file format so that it is easily
accessed across platforms and using a variety of software. Making data
“reusable” includes sharing it with a clear usage license for those who want
to use it in their own work. Recent papers published in _JSDSE_ are made
available under a Creative Commons license, with copyright held by the author.
This gives readers the right to build upon the work. The “accessible” piece of
FAIR Data principles speaks to making data freely available which can broaden
who can do further research on the topic (Nagaraj et al. 2020).

###  4.3 Which repository to use?

Journals like _Nature_ (Nature 2024) and _Proceedings of the National Academy
of Sciences_ (PNAS) (Proceedings of the National Academy of Sciences 2024)
provide detailed guidelines on appropriate repositories for sharing data;
these can be used as references for exploring repository options.

The FAIR Data Principles also provide a framework for determining what makes a
“good” repository. For example, part of the “findable” piece of the FAIR Data
Principles involves having a way to link to the data that is permanent. A
digital object identifier (DOI) is an example of a permanent, FAIR, way to
access a dataset. A GitHub repository is not “findable” in this sense because
the repository can be deleted, renamed, or reorganized, such that the link to
the repository no longer brings a reader to a dataset as intended (FAIRdata
Forum 2021).

_JSDSE_ authors have deposited data in a variety of FAIR repositories. A
number have utilized the OSF (Alzen et al. 2024) which facilitates
anonymization of authors (that can then be unmasked once a paper has been
accepted). Others have used Zenodo (Mocko et al. 2024) and Mendeley Data
(Miller & Pyper 2024), sometimes based on recommendations or requirements of
their institutions.

##  5 Challenges to sharing of deidentified data

Although broad access is a value, the devil really is in the details. There is
a need to balance both transparency and reproducibility with data privacy
(Zaslavsky & Horton 1998). Many ethical considerations exist for data that
often comes from students.

While many investigations might be deemed exempt from human subjects oversight
(involving no or minimal risk), we don’t have to look far to come up with a
plausible, yet, more complicated scenario that our journal has already faced.

Suppose each row in a dataset is a student. There are columns for performance
on a pre-test, performance on a post-test, and a variety of demographic data
including race, major, gender, and class year. If this data were shared
without any names of the students, that would on the surface look
deidentified. However, the combination of demographic variables can reveal
individuals, especially those belonging to intersectional, minority groups.
For example, if there was only one first-year, Asian, male student in the
class, we automatically know how they performed on the pre-test and post-test,
which violates this student’s privacy. (Even if there were more than one such
student, the probability of reidentification is non-zero.)

One option is to remove the sensitive demographic columns of the data, and
focus on the response and covariates of interest. If we are interested in the
relationship between pre-test and post-test scores, the demographic variables
are not essential to reproduce the analysis. Removing potentially identifying
demographic variables in this way and sharing a deidentified dataset that is a
subset of the full data is the approach taken by Evans et al. (2023).

However, if those demographic variables are part of the research question,
i.e., we want to investigate how demographic variables may mediate the
relationship between pre-test and post-test performance, then these variables
would be an important part of the reproducibility process. Another option
might be to create synthetic data (Census Bureau 2021) that has similar
characteristics to the raw data but where each row does not represent an
actual student.

The idea here is that the shared processing and analysis code could be run
using the synthetic data, but no student data would be revealed. Such an
approach would be indicated if there was substantial risk of disclosure
(typically beyond what is considered “exempt from human subjects oversight” by
many IRBs).

##  6 Answers to Frequently Asked Questions

###  6.1 What should a data availability statement look like?

Even in a simple scenario, we can see that each dataset comes with its own
nuance. Therefore, in answer to this question, we resort to a statistician’s
or data scientist’s favorite phrase says: it depends!

Articles published in _JSDSE_ are not intended to adhere to a single, rigid,
data availability policy but rather the intent of requiring a data
availability statement at all is to start a conversation and a creative
problem solving process. The journal wants to work with authors to negotiate a
minimal dataset that balances the dueling goals of accessibility and
protecting the rights of the data subjects.

That being said, we provide several examples of possible data availability
statements that meet the letter and spirit of the guidelines:

  1. 1.

The deidentified data and code that support the findings of this study are
openly available at the Open Science Framework (OSF): provide URL here.

  2. 2.

The authors confirm that the data supporting the findings of this study are
available within the article.

  3. 3.

Data sharing is not applicable to this article as no new data were created or
analyzed in this study.

Student data can come in many forms, their responses to a survey, their
performance on an assessment, their demographics, but they can also come in a
variety of types including qualitative data. Written responses, audio/video,
and interview transcripts are personal and can contain more identifiable
information than quantitative data, which can make sharing the raw data
challenging. Such a study might adopt the following data availability
statement:

  1. 4.

Due to the nature of the research, the video recordings of the participants
are not available. The deidentified coded data and analysis code that support
the findings of this study are openly available at the Open Science Framework
(OSF): provide URL here.

###  6.2 Is share upon request an acceptable option?

Alternative approaches to data availability statements such as “share upon
request” have been notoriously ineffective as a way of fostering data and code
sharing (Tedersoo et al. 2021, Gabelica et al. 2022). “Share upon request” is
not, in general, allowed for _JSDSE_ papers.

###  6.3 How can a waiver be requested?

Authors seeking such an exemption to deidentified data sharing should send a
formal request to the editor asking for a waiver. Such a request should
include (1) the justification for why a deidentified dataset cannot be made
available, (2) background on the study including the original IRB protocol and
study documents, and (3) a codebook for the data included in the waiver.

###  6.4 Where should the data availability statement be located?

In terms of placement, the data availability statement should appear just
before the references. Information about where deidentified data and code can
be found is also provided via metadata input during the submission process. A
stub data availability statement is included in the template (LaTeX and
Quarto) provided for authors (Journal of Statistics and Data Science Education
2024).

###  6.5 Where can I learn more about computational reproducibility?

This editorial has focused most heavily on the details surrounding the sharing
of deidentified data as privacy concerns make additional guidance necessary.
However, the sharing of the code used to process and analyze that data is also
a necessary but insufficient step to foster computational reproducibility
(Project TIER 2024). Sandve et al. (2013) provide ten simple rules for
reproducible computational research. While all are important, rule ten
describes ways to provide public access to scripts and results. Ball et al.
(2022) describe approaches for teaching reproducible science. We encourage
authors to take advantage of the Quarto paper template (Quarto Journals
Project 2024) that has been made available to facilitate reproducible analysis
and reporting.

##  7 Closing thoughts

We realize that best practices to foster reproducibility and replicability are
fast changing and require additional efforts by authors, reviewers, and
editors.

This process will require updates to our standard procedures, including
improved templates for human subjects oversight. For example, these will need
to no longer state that “deidentified data will be destroyed at the end of the
study”. It will be equally important to inform subjects that fully
deidentified data will be made available and not just summarized in aggregate.

However, we believe that these changes can and should be undertaken, and that
these changes will allow us as a community to balance privacy and sharing in a
way that fosters better science.

At present, the deidentified data and code included with submissions are not
formally included in the review process (though some reviewers and associate
editors may choose to incorporate this information). In the future, the
journal may consider formalizing this review and/or starting up a
reproducibility process such as that described by Wrobel et al. (2024) or the
creative approach undertaken by the journals published by the American
Economic Association (Vilhuber et al. 2022).

We hope that this discussion will benefit future authors and readers as they
navigate this new territory. We concur with the assessment of Wrobel et al.
(2024) which stated that “the journal gains credibility from a system that
holds researchers accountable for their work, credibility that can be
leveraged to publish controversial and impactful research that has the
potential to change a field of study”.

Nicholas J. Horton ORCID: https://orcid.org/0000-0003-3332-4311

Sara Stoudt ORCID: https://orcid.org/0000-0002-1693-8058

##  8 Acknowledgements

We acknowledge helpful comments on an earlier draft of the manuscript from
Richard Ball, Johanna Hardin, Chris Paciorek, Joshua Rosenberg, and Jeffrey
Witmer.

## Disclosure statement

The authors have no conflicts of interest to declare.

## Data availability statement

No data or code are associated with this editorial.

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