On some integrals of Hardy
Alexander E Patkowski
We consider some properties of integrals considered by Hardy and Koshliakov, and which have also been further extended recently by Dixit. We establish a new general integral formula from some observations about the digamma function. We also obtain lower and upper bounds for Hardy's integral through properties of the digamma function.
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\begin{document}
\author{Alexander E Patkowski}
\title{On some integrals of Hardy}

\maketitle
\begin{abstract}We consider some properties of integrals considered by Hardy and Koshliakov, and which have also been further extended recently by Dixit. We establish a new general integral formula from some observations about the digamma function. We also obtain lower and upper bounds for Hardy's integral through properties of the digamma function. \end{abstract}

% AMS keywords (used in AMS journals)
\keywords{\it Keywords: \rm Fourier Integrals; Riemann xi function; Digamma function.}

% AMS subject classifications (used in AMS journals)
\subjclass{ \it 2010 Mathematics Subject Classification 11M06, 33C15.}

\section{Introduction and Main formulas}
In a paper written by the well-known G. H. Hardy [10], an interesting integral formula is presented (corrected in [3])
\begin{equation} \int_{0}^{\infty}\frac{\Xi(t/2)}{1+t^2}\frac{\cos(xt)}{\cosh(\pi t/2)}dt=\frac{1}{4}e^{-x}\left(2x+\frac{1}{2}\gamma+\frac{1}{2}\log(\pi)+\log(2)\right)+\frac{1}{2}e^{x}\int_{0}^{\infty}\psi(t+1)e^{-\pi t^2e^{4x}}dt,\end{equation}
where $\psi(x):=\frac{\partial}{\partial x}\log(\Gamma(x)),$ $\Gamma(x)$ being the gamma function [1], and $\Xi(t):=\xi(\frac{1}{2}+it),$ where [11, 15]
$$\xi(s):=\frac{1}{2}s(s-1)\pi^{-\frac{s}{2}}\Gamma(\frac{s}{2})\zeta(s).$$ Here we have used the standard notation for the Riemann zeta function $\zeta(s):=\sum_{n\ge1}n^{-s},$ for $\Re(s)>1.$ Koshliakov [12] produced this formula as well, but in a slightly different form, using instead the integral 
\begin{equation}e^{x}\int_{0}^{\infty}(\psi(t+1)-\log(t))e^{-\pi t^2e^{4x}}dt,\end{equation}
for the right side of (1.1).
Here the trick, which has been exploited in the many studies [3, 4, 5, 6, 7, 12, 14], is to re-write the left side of (1.1) as an inverse Mellin transform by utilizing the classical functional equation $\xi(s)=\xi(1-s)$ [15]. Titchmarsh has used simpler but similar integrals as the left side of (1.1) to obtain Hardy's result that $\Xi(t)$ has infinitely many real zeros [15]. See Csordas [2] for some more recent work in this direction. A recent paper by Dixit offers a beautiful generalization of formula (1.1), which involves a confluent hypergeometric function [3]. Other refinements can be found in [4, 5]. \par The purpose of this paper is to offer some further results concerning (1.1) that appear to have been overlooked. A new generalization is offered as well that takes a different route than the ones offered by [3, 4, 5], considering what is essentially the polygamma function [1]. In the following section, we offer some new inequalities that we noticed upon examining the integral given in (1.1).
\begin{theorem} For $m\in\mathbb{N}_{0},$ we have that 
$$(-1)^{m}e^{x}\int_{0}^{\infty}t^m \bar{\psi}_{m}(t)e^{-\pi t^2e^{4x}}dt=\frac{1}{2}\int_{0}^{\infty}\frac{\Xi(t)}{\cosh(\pi t)(t^2+\frac{1}{4})}\left(\frac{\Gamma(\frac{1}{2}-it+m)}{\Gamma(\frac{1}{2}-it)}e^{-2xit}+\frac{\Gamma(\frac{1}{2}+it+m)}{\Gamma(\frac{1}{2}+it)}e^{2ixt}\right)dt,$$
$$\bar{\psi}_{m}(t):=\frac{\partial^m }{\partial t^m}(\psi(t+1)-\log(t)).$$

\end{theorem}
\begin{proof} Recall the Mellin transform of a suitable function $f(t)$ is given by [1]
$$\int_{0}^{\infty}f(t)t^{s-1}dt=F(s),$$ and its inverse,
$$f(t)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}F(s)t^{-s}ds,$$
provided $c$ is a real number chosen where $F(s)$ is analytic.
This is a special application of Parseval's theorem for Mellin transforms with one function chosen as $f(t)=e^{-\pi t^2},$ and the other as $g(t)=t^m \bar{\psi}_{m}(t).$ From Titchmarsh [15, eq.(2.15.7)] we have for $0<c<1,$ 
\begin{equation}\psi(x+1)-\log(x)=-\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\frac{\pi\zeta(1-s)}{\sin(\pi s)}x^{-s}ds.\end{equation}
Differentiating (1.3) $m$ times, gives us
\begin{equation}\bar{\psi}_{m}(x)=(-1)^{m-1}\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\frac{\pi\zeta(1-s)}{\sin(\pi s)}\frac{\Gamma(s+m)}{\Gamma(s)}x^{-s-m}ds.\end{equation}
Now proceeding as we mentioned in the beginning in the proof, noting $\sin((\frac{1}{2}+it)\pi)=\cosh(\pi t),$ the functional equation $\xi(s)=\xi(1-s),$ the Mellin transform of $e^{-\pi(xt)^2}$ is $\pi^{-s/2}x^{-s}\Gamma(s/2)/2,$ with $0<c'<1$
$$\int_{0}^{\infty}f(t)g(t)dt=\frac{1}{2\pi i}\int_{c'-i\infty}^{c'+i\infty}F(s)G(1-s)ds,$$
we get $0<c'<1$
$$\frac{1}{2\pi i}\int_{c'-i\infty}^{c'+i\infty}\frac{\pi\xi(s)}{\sin(\pi s)}\frac{\Gamma(1-s+m)}{\Gamma(1-s)}x^{-s}ds.$$
Put $c'=\frac{1}{2}$ and we find this is equal to
$$\frac{1}{2\sqrt{x}}\int_{-\infty}^{\infty}\frac{\Xi(t)}{\cosh{\pi t}}\frac{\Gamma(\frac{1}{2}-it+m)}{\Gamma(\frac{1}{2}-it)}x^{-it}dt.$$
Splitting this bilateral integral up into two integrals and replacing $x$ by $e^{2x}$ leads to the result.


\end{proof}

Koshliakov [12] gives a similar integral to (1.1) but squaring the $\Xi(t)(t^2+\frac{1}{4})^{-1}$ function. Moll and Dixit state his result in [6, Theorem 4.4], and offer further interesting generalizations. However, we noticed a slightly different form of this integral based on the type of proof we used in the previous theorem. 

\begin{theorem} For $x>0,$
\begin{equation}\int_{0}^{\infty} (\psi(t+1)-\log(t))\left(\frac{2\pi e^{x/2}}{t}-e^{-x/2}4\sum_{n\ge1}K_{0}(2\pi nte^{-x})\right)dt=\int_{0}^{\infty}\frac{\Xi^2(t)}{(t^2+\frac{1}{4})^2}\frac{\cos(xt)}{\cosh(\pi t)}dt.\end{equation}
\end{theorem}

\begin{proof} First, since $K_{0}(t)=O(t^{-1/2}e^{-t}),$ for $t\rightarrow\infty$ in $|\arg(t)|<3\pi/2$ where as usual $K_s(t)$ is the modified Bessel function, it follows that the 
series on the right side of the formula [9] $d>0,$
$$\sum_{n\ge1}\frac{1}{2\pi i}\int_{d-i\infty}^{d+i\infty}\Gamma^2(\frac{s}{2})(\frac{x}{2n})^{-s}ds=4\sum_{n\ge1}K_{0}(n\pi x),$$
converges absolutely. Summing through, we find for $d'>1,$
$$\frac{1}{2\pi i}\int_{d'-i\infty}^{d'+i\infty}\Gamma^2(\frac{s}{2})(\frac{x}{2})^{-s}\zeta(s)ds=4\sum_{n\ge1}K_{0}(n\pi x),$$
which has a simple pole at $s=1.$ If we calculate this residue and move the line of integration to $\Re(s)=b,$ $0<b<1,$ we find
\begin{equation}\frac{1}{2\pi i}\int_{b-i\infty}^{b+i\infty}\Gamma^2(\frac{s}{2})(\frac{x}{2})^{-s}\zeta(s)ds+\frac{2\pi}{x}=4\sum_{n\ge1}K_{0}(n\pi x).\end{equation}
Now noting (1.3) and (1.6) in conjunction with Parseval's theorem for Mellin transforms, we obtain the result after applying the functional equation for $\xi(s).$ 
\end{proof}
Other associated integral formulae related to Fourier cosine transforms may be obtained from using known evaluations.
\begin{theorem} We have for $\frac{\pi}{2}>|\Im(\beta')|,$
\begin{equation}\int_{0}^{\infty}\frac{\cosh(t)}{\cosh(2t)+\cosh(2\beta')}\left(e^{t/2}-2e^{-t/2}\sum_{n\ge1}e^{-\pi n^2e^{-2t}}\right)dt\end{equation}
$$=\frac{\pi e^{\beta'}}{4\cosh(\beta')}\int_{0}^{\infty}(\psi(x+1)-\log(x))e^{-\pi x^2e^{4\beta'}}dx.$$
\end{theorem}
\begin{proof} We apply the classical [15, eq.(2.16.2)]
$$\int_{0}^{\infty}\frac{\Xi(t)}{(t^2+\frac{1}{4})}\cos(xt)dt=\frac{\pi}{2}(e^{t/2}-2e^{-t/2}\sum_{n\ge1}e^{-\pi n^2e^{-2t}}),$$
with the formula [9], valid for $\pi\Re(\alpha)>|\Im(\bar{\alpha}\beta)|,$ and
$$\int_{0}^{\infty}\frac{\cos(ty)\cosh(t\alpha/2)dt}{\cosh(\alpha t)+\cosh(\beta)}=\frac{\pi}{2\alpha}\frac{\cos(\beta y/\alpha)}{\cosh(\beta/2)\cosh(\pi y/\alpha)}.$$
We need only let $\beta\rightarrow2\beta',$ and $\alpha=2,$ and then we have $\frac{\pi}{2}>|\Im(\beta')|.$ This coupled with Parseval's theorem gives the result.

\end{proof}
Note that the integral on the left side of (1.6) hints to us about the non-negativity of the integral on the right side. We will investigate this further in the next section.

\section{Inequalities}  We first start with some preliminaries about some special functions which can be found in [1]. First we recall the error function,
$$\erf(x)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}e^{-t^2}dt,$$
 and its complement, $\erfc(x)=1-\erf(x).$ Also, $\erfi(x):=-i\erf(ix).$ We also use the exponential integral
 $$\Ei(x)=-\int_{-x}^{\infty}\frac{e^{-t}}{t}dt. $$
\begin{theorem} Define the Hardy integral for $y>0$ as
$I(y):=\int_{0}^{\infty}(\psi(t+1)-\log(t))e^{-yt^2}dt.$ Then we have,
$$\frac{(2c_1)^2\sqrt{y}}{\sqrt{\pi}\erfi(\sqrt{y})}-\frac{\Ei(-y)}{4}-\frac{e^{-y}}{12}+\frac{\sqrt{y\pi}}{12}\erfi(\sqrt{y})\le I(y).$$
Furthermore,
$$I(y)\le \sqrt{c_2\sqrt{\frac{\pi}{2y}}\erf(\sqrt{2y})}-\frac{\Ei(-y)}{4},$$
where $c_1=0.952894,$ and $c_2=1.56624.$
\end{theorem}

\begin{proof} We first need a result from the paper [13, Corollary 1], for $x>0,$
\begin{equation}\frac{1}{2x}-\frac{1}{12x^2}<\psi(x+1)-\log(x)<\frac{1}{2x}.\end{equation}
For the upper bound we first write,
$$I(y)=\int_{0}^{1}(\psi(t+1)-\log(t))e^{-yt^2}dt+\int_{1}^{\infty}(\psi(t+1)-\log(t))e^{-yt^2}dt.$$
Applying (2.1) we have that this is
$$\le \int_{0}^{1}(\psi(t+1)-\log(t))e^{-yt^2}dt-\frac{\Ei(-y)}{4}.$$
By Schwarz's inequality and the fact that 
$$\int_{0}^{1}(\psi(t+1)-\log(t))^2dt=1.56624$$ we have the upper bound. For the lower bound, we write the Schwarz's inequality in the form
$$\left(\int_{0}^{1}(\psi(t+1)-\log(t))^{1/2}dt\right)^2\le\left(\int_{0}^{1}(\psi(t+1)-\log(t))e^{-yt^2}dt\right)\left(\int_{0}^{1}e^{yt^2}dt\right),$$
and also note,
$$\int_{0}^{1}\sqrt{(\psi(t+1)-\log(t))}dt=0.952894.$$ This coupled with similar computations derived from the left side of (2.1), together with the definitions of the special integral functions, gives our result.
\end{proof}

Now it can be seem that the lower bound actually implies the Hardy integral $I(x)$ is non-negative for $x>0,$ and also the squeeze theorem shows that $I(x)\rightarrow0$ when $x\rightarrow\infty.$ 

\section{Other observations and further comments}
While studies such as this one and others have shown there is a plethora of integral formulae like Hardy's (1.1) to be found, it would be interesting to see if these integrals could be incorporated into similar work like Csordas [2]. We were interested in seeing if (1.1) could be adapted to imply information beyond Hardy's theorem, that $\Xi(t)$ has infinitely many real zeros. The classical ideas are included in Titchmarsh's text [15, pg. 256-260], and evolve around the ideas of first assuming $\Xi(t)$ is of one sign, and then arriving at a contradiction from properties of an integral of the form $\int_{0}^{\infty}\Xi(t)k(t)dt,$ $k(t)$ non-negative for $t>0.$ The idea in [15, pg. 260] considers the moments $k(t)=t^{2n}.$ Apply the operator $\bar{\partial}_x=-\partial_x^2+\frac{1}{4}$ of Glasser [8] to (1.1) (after replacing $t$ by $t/2$) and set $x=x'+i\pi$ to get
$$\lim_{x\rightarrow x'+i\pi}\bar{\partial}_x(e^xI(\pi e^{4x}))=\int_{0}^{\infty}\Xi(t)\cos(x't)dt+i\int_{0}^{\infty}\Xi(t)\frac{\sin(x't)\sinh(\pi t)}{\cosh(\pi t)}dt.$$
Differentiating $2n$ times and setting $x'=0$ gives us the right side as the value $(-1)^n\int_{0}^{\infty}t^{2n}\Xi(t)dt.$ Note that $e^{x'+i\pi}I(\pi e^{4(x'+i\pi)})=-e^{x'}I(\pi e^{4x'}).$ Further the integral $I(x)$ clearly converges only for $\Re(x)>0,$ and hence $I(\pi e^{4z})$ converges when $-\frac{\pi}{4}<\Im(z)<\frac{\pi}{4},$ and $\lim_{z\rightarrow i\pi/4}I(\pi e^{4z})=+\infty.$ It would be interesting if new information could be obtained about $\Xi(t)$ from alternative forms of these moments like those found here.


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\end{thebibliography}
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E-mail: alexpatk@hotmail.com
\end{document}

1702.05635.extracted_bib
{ConcreteMath G. Andrews, R. Askey, and R. Roy. Special Functions, volume 71 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, New York, 1999.
1702.05635.extracted_bib
{ConcreteMath G. Csordas, Fourier Transforms of Positive Definite Kernels and the Riemann $\xi$-function, Computational Methods and Function Theory, Volume 15, Issue 3, pp 373--391 (2015).
1702.05635.extracted_bib
{ContreteMath A. Dixit. Analogues of the general theta transformation formula, Proc. Roy. Soc. Edinburgh, Sect. A, 143:371--399, 2013.
1702.05635.extracted_bib
{ConcreteMath A. Dixit, N. Robles, A. Roy and A. Zaharescu, Zeros of combinations of the Riemann $\xi$-function on bounded vertical shifts, J. Number Theory Volume 149, April 2015, Pages 404?434
1702.05635.extracted_bib
{ConcreteMath A. Dixit, A. Roy and A. Zaharescu, Riesz-type criteria and theta transformation analogues, J. Number Theory 160, p. 385--408 (2016).
1702.05635.extracted_bib
{ConcreteMath A. Dixit, V. H. Moll Self-reciprocal functions, powers of the Riemann zeta function and modular-type transformations, J. Number Theory 147 (2015), 211--249.
1702.05635.extracted_bib
{ConcreteMath W. L. Ferrar, Some solutions of the equation $F(t) = F(t^{-1)$, J. London Math. Soc. 11 (1936), 99--103.
1702.05635.extracted_bib
{ConcreteMath M.L. Glasser A note on the Fourier transform, Journal of Physics A: Mathematical and Theoretical, Volume 49, Number 45 (2016), 454004.
1702.05635.extracted_bib
{ConcreteMath I. S. Gradshteyn and I. M. Ryzhik. Table of Integrals, Series, and Products. Edited by A.Jeffrey and D. Zwillinger. Academic Press, New York, 7th edition, 2007.
1702.05635.extracted_bib
{ConcreteMath G.H. Hardy, Note by Mr. G.H. Hardy on the preceding paper, Quart. J. Math. 46 (1915), 260--261.
1702.05635.extracted_bib
{ConcreteMath H. Iwaniec and E. Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004.
1702.05635.extracted_bib
{ConcreteMath N. S. Koshlyakov, On a transformation of definite integrals and its application to the theory of Riemann's function $\zeta(s)$, Comp. Rend. (Doklady) Acad. Sci. URSS 15 (1937), 3--8.
1702.05635.extracted_bib
{ConcreteMath Feng Qi, Run-Qing Cui, Chao-Ping Chen, Bai-Ni Guo Some completely monotonic functions involving polygamma functions and an application, J. Math. Anal. Appl. 310 (2005) 303--308.
1702.05635.extracted_bib
{ConcreteMath S. Ramanujan, New expressions for Riemann's functions $\xi(s)$ and $\Xi(t)$, Quart. J. Math., 46: 253--260, 1915.
1702.05635.extracted_bib
{ConcereteMath E. C. Titchmarsh, The theory of the Riemann zeta function, Oxford University Press, 2nd edition, 1986.