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This forum brings together a broad enough base of mathematicians to collect a "big list" of equivalent forms of the Riemann Hypothesis...just for fun. Also, perhaps, this collection could include statements that imply RH or its negation.

Here is what I am suggesting we do:

Construct a more or less complete list of sufficiently diverse known reformulations of the Riemann Hypothesis and of statements that would resolve the Riemann Hypothesis.

Since it is in bad taste to directly attack RH, let me provide some rationale for suggesting this:

1) The resolution of RH is most likely to require a new point of view or a powerful new approach. It would serve us to collect existing attempts/perspectives in a single place in order to reveal new perspectives.

2) Perhaps the resolution of RH will need ideas from many areas of mathematics. One hopes that the solution of this problem will exemplify the unity of mathematics, and so it is of interest to see very diverse statements of RH in one place. Even in the event where no solution is near after this effort, the resulting compilation would itself help illustrate the depth of RH.

3) It would take very little effort for an expert in a given area to post a favorite known reformulation of RH whose statement is in the language of his area. Therefore, with very little effort, we could have access to many different points of view. This would be a case of many hands making light work. (OK, I guess not such light work!)

Anyhow, in case this indeed turns out to be an appropriate forum for such a collection, you should try to include proper references for any reformulation you include.

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    $\begingroup$ You can find a few at: en.wikipedia.org/wiki/Riemann_hypothesis $\endgroup$ – S. Carnahan Sep 25 '10 at 12:41
  • $\begingroup$ There was an AIM Problem List which included a bunch of examples in connection with last year's "RH Day." Although the link: aimpl.org/pl seems not to be working anymore, at least there is this Archive.Org link: web.archive.org/web/20120731034246/http://aimath.org/pl/… (hat tip to John Washburn). $\endgroup$ – Micah Milinovich Sep 25 '10 at 15:15
  • $\begingroup$ I think this is a great question and have more than once wished for such a list in the past. $\endgroup$ – Louigi Addario-Berry Sep 25 '10 at 20:04
  • $\begingroup$ another benefit of this question is that it helps people who are not as familiar with RH. For example, I may understand some of the equivalent formulations a bit better than the original formulation. $\endgroup$ – Sean Tilson Sep 25 '10 at 21:06
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    $\begingroup$ Maybe people should vote for what they think is currently the most promising approach, based on an equivalent reformulation $\endgroup$ – Alex R. Sep 25 '10 at 23:47

21 Answers 21

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I like Lagarias "elementary" reformulation of Robin's theorem: that RH is true iff

$\sigma(n)\leq H_n+e^{H_n}\log(H_n)$

holds for every $n\geq 1$, where $\sigma(n)$ is the sum of divisors function and $H_n$ is the nth harmonic number.

Its major appeal is that anyone with rudimentary exposure to number theory can play with it. Having spent the better part of my youth fiddling with this reformulation really brought out the enormous difficulty of proving RH. In a way I think this reformulation is evil, because it looks tractable, but is ultimately useless and perhaps even harder to work with than other more complex reformulations. On the other hand I hope a future proof of RH will involve this reformulation because then I might have a chance of understanding the proof!

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    $\begingroup$ I used to teach a computer algebra class, and at the exam I asked the student to write a program checking this inequality on random examples. The possibility of winning a million dollars if they find a counter example (minus my share, of course) is a nice incentive :). $\endgroup$ – Adrien Aug 3 '18 at 12:16
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    $\begingroup$ Where can I find the proof that it is equivalent to the originial RH statement? $\endgroup$ – NewBornMATH Apr 17 at 3:54
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    $\begingroup$ @NewBornMATH: math.lsa.umich.edu/~lagarias/doc/elementaryrh.pdf $\endgroup$ – Alex R. Apr 17 at 7:06
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In the article Seized opportunities (Notices of the AMS, April 2010), Victor Moll gives the following, which he credits to V.V.Volchkov. Establishing the exact value $$\int_{0}^{\infty}\frac{(1-12t^2)}{(1+4t^2)^3}\int_{1/2}^{\infty}\log|\zeta(\sigma+it)|~d\sigma ~dt=\frac{\pi(3-\gamma)}{32}$$ is equivalent to the Riemann Hypothesis. Moll cheekily adds that evaluating that integral might be hard.

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    $\begingroup$ Oh, and $\gamma$ is Euler-Mascheroni. $\endgroup$ – Koundinya Vajjha Mar 9 '11 at 13:07
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This one is not too bad though not big:

Equivalences to the Riemann Hypothesis, ed. J. Brian Conrey and David W. Farmer

Yet there are many (above a hundred at least) and it depends on the type you are looking for. Analytic elementary number theory ....

ADDED LATER : My favorite is very elementary:
Among the square free integers below $N$:
Let $D(N)$ denote the absolute value of the difference between the number of those divisible by an even number of primes and the number of those divisible by an odd number of primes .

R.H. says that $D(N)$ comes close to the square root of $N$.

More precisely: for any $\epsilon > 0 $ there is $N_0$ such that any $N > N_0$ verifies $ {D(N)} \leq N^{1/2+\epsilon}$.

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Robin's criterion has been written in various places in MO: define Gronwall's function $$ G(n) = \frac{\sigma(n)}{n \log \log n}. $$ In 1984, Robin showed that RH is equivalent to $$ G(n) < e^\gamma, \; \; \forall n \geq 5041.$$

Robin's adviser was Jean-Louis Nicolas. There is a new equivalence due to Nicolas, G. Caveney, and J. Sondow. Define a positive integer $N$ to be $GA1$ if $N$ is composite and $G(N) \geq G(N/p)$ for all primes $p |N.$ Let $N$ be called $GA2$ if $G(N) \geq G(aN)$ for all positive integers $a,$ where in this case we allow $N$ to be prime or composite. Then RH is equivalent to the assertion that the only number that is both $GA1$ and $GA2$ is 4. See arXiv and arXiv

I learned of this because Sondow wrote to me asking for a pdf of Robin 1984. And I wrote back. Which people ought to do.

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  • $\begingroup$ Since I am still worried you never saw my reply, I will seize this opportunity to thank you again for your kind offer to send me this paper when we both answered a question related to Robin's criterion this January. (I only replied via a comment with a delay of a couple days as I was off-line; and only via a comment as I did not need the paper sended.) In any case, thanks again for the offer! $\endgroup$ – user9072 Aug 22 '12 at 12:46
  • $\begingroup$ @quid, I suspect I never saw that comment. I have noticed that there are some bugs in the comment notification system. Even though people correctly write after an answer of mine, or add @Will when the comments are at some other location, i do not always get notified (the little envelope at top turning orange). I generally check my own activity from the day before to see for changes or comments, but if an extra day passed I might not have thought to check any more. $\endgroup$ – Will Jagy Aug 22 '12 at 18:19
  • $\begingroup$ Then good I wrote this, like, better late than never. Just one tangential point: the @name notification does not work here 'by design'. This feature was only added in newer version of SE than currently in use here (but after the move it should work); and the thing was a reply to your comment on my answer, so there was no notification by design. $\endgroup$ – user9072 Aug 22 '12 at 18:45
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    $\begingroup$ @quid, found it, mathoverflow.net/questions/84266/on-robins-criterion-for-rh $\endgroup$ – Will Jagy Aug 22 '12 at 19:37
  • $\begingroup$ Is it expected that $G(n)$ approaches $e^\gamma$ or may be $G(n)$ can be $1$ or any small constant as desired? $\endgroup$ – T.... Jul 31 '17 at 11:42
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The following is given without source here:

RH is equivalent to the assertion that for all $n\ge3$ $$| \log \operatorname{lcm}(1,2,\dots, n) - n | < \sqrt{n}\log^2(n)$$ where $\operatorname{lcm}$ denotes the least common multiple.

More details about this function (also called second Chebyshev function) can be found in this Wikipedia entry (thank you Wojowu!).

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  • $\begingroup$ Very interesting. Is it anyhow related to the conjectural upper bound for the quantity $\alpha_{n}$ defined in mathoverflow.net/questions/61842/about-goldbachs-conjecture ? $\endgroup$ – Sylvain JULIEN Feb 29 '16 at 13:56
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    $\begingroup$ For the record, the function $\log lcm(1,2,\dots,n)$ is equal to second Chebyshev function $\psi(n)$. $\endgroup$ – Wojowu Feb 29 '16 at 17:25
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Lapidus and Maier show that “One can hear the shape of a fractal string of dimension $D \neq \frac12$” if and only if the Riemann hypothesis is true.

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A good resource is The Riemann Hypothesis by Borwein, Choi, Rooney, and Weirathmueller, CMS, 2008. It has equivalences plus much more.

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Not especially sophisticated, but there is the note by Tuck: When does the first derivative exceed the geometric mean of a function and its second derivative?. This is studied some more in

Tuck's incompressibility function: statistics for zeta zeros and eigenvalues
M V Berry and P Shukla 2008 J. Phys. A: Math. Theor. 41 385202

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Found in this question

DMR 1974: http://books.google.ca/books?id=4lT3M6F745sC&pg=PA335

$$\eta(j)=p \text{ if } j=p^k, \; p \text{ is prime}$$ $$\eta(j)=1 \text{ otherwise}$$ $$ \delta(x)=\prod_{n < x }\prod_{ j \le n} \eta(j) $$

RH is equivalent to the assertion that $$\left( \sum_{k \le \delta(n)}\frac1k - \frac{n^2}{2}\right)^2 < 36n^3$$

for $n \ge 1$.

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  • $\begingroup$ Since it may be hard to look inside that Google Book, let me mention that DMR 1974 stands for Martin Davis, Yuri Matiyasevic, and Julia Robinson (1974), Hilbert's tenth problem. Diophantine equations: positive aspects of a negative solution. I like joro's statement for its being so nakedly expressible in Peano arithmetic (there are others of course, but this one is most obviously so expressible). $\endgroup$ – Todd Trimble Mar 25 '18 at 16:31
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I met a guy today who I convinced to sign up on MathOverflow. His name is Kevin Broughan and he has a couple of volumes on this subject, divided into arithmetic and analytic equivalents. You might check it out.

Gerhard "Oh, The People You Meet...." Paseman, 2018.08.02.

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Two statements I find quite interesting involve Farey sequences, and, very roughly, they can be interpreted as saying that elements of Farey sequence are "not far" from being evenly distributed in the unit interval.

To be precise, suppose $n$-th Farey sequence is $0=a_{0,n}<a_{1,n}<\dots<a_{m_n,n}=1$ and let $d_{k,n}=a_{k,n}-\frac{k}{m_n}$. Then RH is equivalent to any of the two statements below:

  • $\sum_{k=0}^{m_n} d_{k,n}^2=O(n^{-1+\varepsilon})$ for any $\varepsilon>0$ as $n\rightarrow\infty$.
  • $\sum_{k=0}^{m_n} |d_{k,n}|=O(n^{1/2+\varepsilon})$ for any $\varepsilon>0$ as $n\rightarrow\infty$.
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I can't seem to find it in any of the answers here, so:

In three papers, Roesler presents an $n\times n$ matrix $\mathbf A_n$ whose determinant is $n! \sum_k \frac{\mu(k)}{k}$, (from which the statement that is equivalent to the hypothesis is $\det \mathbf A_n=O(n! n^{\varepsilon-\frac12})$ for all positive $\varepsilon$). He then proceeds to study the eigenvalues of this matrix to glean insights into the hypothesis from this viewpoint.

Relatedly, this paper by Barrett and Jarvis deals with a matrix originally studied by Redheffer, whose determinant is the Mertens (summatory Möbius) function $\sum_k\mu(k)$. They also then study the eigenvalues of this matrix.

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there are many matrix formulations of the RH. this one is my favorite : a sequence of symmetric matrices related to the Mertens function.

one consider the set $$S_n = \left\{ s \in \mathbb{N}\ \left| \ \ s = \left\lfloor \frac{n}{k}\right\rfloor , \ k \in \mathbb{N} \right.\right\}$$ and the $|S_n| \times |S_n|$ matrix $A^{(n)}_{ \ \ ij} = \text{Mertens}\left(\left\lfloor \frac{n}{s_i s_j} \right\rfloor\right)$.

given that $\text{Mertens}(n) \le \|A^{(n)}\| = \max_{\|u\|=1} \|A^{(n)} u\|$ , the Riemann hypothesis would be implied by $$\|A^{(n)}\| = \mathcal{O}(n^{1/2+\epsilon})$$

what is amazing is that the sequence $\|A^{(n)}\|$ is one of the smoothest you'll ever see being related to the Riemann hypothesis, and the fact that $\text{Mertens}(n)$ only depends on the $|S_n|{\scriptstyle-}1 < 2\sqrt{n}$ previous values of $\text{Mertens}\left(s_i\right)$ which is perfectly encoded in that matrix sequence.

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By Value Distribution Theory Related to Number Theory, Riemann's Hypothesis is true if and only if $$\frac{1}{\pi}\int_0^{\infty} \log\left|\frac{\zeta(\frac{1}{2}+it)}{\zeta(\frac{1}{2})}\right|\ \frac{dt}{t^2}=\frac{\pi}{8}+\frac{\gamma}{4}+\frac{\log 8\pi}{4}-2$$

And a more general theorem has be proved in this book:

Take $a\in R$ with $\frac{1}{2}\leq a<1$. Riemann's $\zeta$-function has no zeros in $\Re(s)>a$ if and only if $$\frac{1}{\pi}\int_0^{\infty} \log\left|\frac{\zeta(a+it)}{\zeta(a)}\right|\ \frac{dt}{t^2}=\frac{\zeta'(a)}{2\zeta(a)}-\frac{1}{1-a}$$

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Two other articles, FWIW:

http://arxiv.org/pdf/0808.0640.pdf (mainly a criteria of Riesz and Baez-Duarte)
http://arxiv.org/pdf/1003.3392.pdf (introducing the "Acid zeta function")

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Let $C_{c}^{r}(\mathbb{R}^{\ast})$ denote the set of all functions $f \colon (0,\infty) \to \mathbb{C}$ that are $r$ times differentiable and have compact support. For any $y \in (0,\infty)$ and any $f \in C_{c}^{2}(\mathbb{R}^{\ast})$, let $m_{y}(f) := \sum_{n \in \mathbb{N}} y\phi(n) f(y^{\frac{1}{2}}n)$ where $\phi$ is the Euler totient function; in addition, let $m_{0}(y) := \int_{0}^{\infty} \left(\frac{6}{\pi^{2}}\right)u f(u)\, du$.

In the paper Discrete measures and the Riemann hypothesis (Kodai Math. J. 17 (1994), no. 3, 596–608.), Prof. Alberto Verjovsky proved the folllowing:

The Riemann Hypothesis is true if and only if $m_{y}(f) = m_{0}(f) + o(y^{\frac{3}{4}-\epsilon})$ as $y \to 0$ for every $f \in C_{c}^{r}(\mathbb{R}^{\ast})$ with $r \in [2,\infty]$ and every $\epsilon > 0$.

You can find the aforementioned paper here: http://projecteuclid.org/euclid.kmj/1138040054

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    $\begingroup$ This was proven much earlier by Peter Sarnak ("Asymptotic behavior of periodic orbits of the horocycle flow and eisenstein series") based on prior work of Zagier (or at-least so Peter said to me once). $\endgroup$ – Asaf Mar 7 '16 at 5:55
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    $\begingroup$ Didn't know that... In any case, it seems to me that neither of the said papers is well-known. $\endgroup$ – José Hdz. Stgo. Mar 8 '16 at 21:26
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    $\begingroup$ Sarnak's paper is very well-known in the field (and its "younger cousin" - the Sarnak-Ubis paper which is relatively recent, and related works by Strombergsson). The reason why it cited so few times is that qualitatively the result (without the RH connection) can be found from the Margulis' mixing argument (from his thesis), which is extremely famous and useful and its a bit more general, the problem with Margulis' approach is that at-best-case you'll hit the Selberg-Ramanujan conjecture (if you generalize the result in the obvious manner for general congruence surface), see Burger's article. $\endgroup$ – Asaf Mar 8 '16 at 21:59
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    $\begingroup$ The beauty of Sarnak's paper is that it goes deeper from Ramanujan, just because of explicitly analyzing the location of the poles of the Eisenstein series.Those quantities (and in general,the(constant term of the-)Eisenstein series for congruence surfaces) will include the Zeta function and then is when you get RH, so the "purely dynamical approach" won't give you RH in any case, but it might recover Ramanujan (related work is Eisendler-Margulis-Venkatesh where they recovered most of Clozel property(Tau) result). anyhow,the reference should be Sarnak's or Zagier's work,and not Verjovsky's $\endgroup$ – Asaf Mar 8 '16 at 22:04
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On the arXiv this morning, The Landau function and the Riemann Hypothesis by Marc Deleglise and Jean-Louis Nicolas:

The Landau function $g(n)$ is the maximal order of an element of the symmetric group of degree $n$; it is also the largest product of powers of primes whose sum is $\le n$. The main result of this article is that the property

For all $n > 0$ , $\log g(n) < li^{-1} (n))$

(where $li^{-1}(n)$ denotes the inverse function of the logarithmic integral) is equivalent to the Riemann hypothesis.

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  • $\begingroup$ Something very like this was posted (but then deleted) by user68208 in March 2016. $\endgroup$ – Gerry Myerson Jul 18 at 23:22
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Using Corollary 1 in Schoenfeld's 1976 paper "Sharper bounds for the Chebyshev functions $\theta(x)$ and $\psi(x)$. II", we see with a bit of numeric work that the Riemann Hypothesis is equivalent to the following inequality: $$|\pi(x)-\mathrm{li}(x)|<\sqrt{x}\log x,\qquad x\geq 2.$$

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Polya-Hurwitz program.

This may become more interesting in light of the recent progress in the Polya-Jensen program by Griffin, Ono, Rolen, Zagier.

We will first provide definitions of some functions involved.

The Riemann Xi-function $\Xi(z)$ is related to the Riemann zeta-function $\zeta(s)$ via ([A], [B]): $\Xi(z)=\xi(\tfrac{1}{2}+iz)$, $\xi(s)=\tfrac{1}{2}s(s-1)\pi^{-s/2}\Gamma\left(\frac{s}{2}\right)\zeta(s)$.

Riemann $\Xi(z)$ can be expressed as a Fourier transform of a positive, fast decaying, and even kernel $\Phi(t)$ ([A], [B]):

\begin{equation} \Xi(z)=2\int_0^{\infty}\Phi(t)\cos(zt)\mathrm{d}t,\tag{1} \end{equation}

where

\begin{equation} \Phi(t)=\sum_{k\geqslant 1}\phi_k(t)=\Phi(-t),\tag{2}\\ \end{equation}

\begin{equation} \phi_k(t)=\left(4\pi^2 k^4 e^{9t/2}-6\pi k^2e^{5t/2}\right)\exp\left(-\pi k^2 e^{2t}\right)\tag{3}. \end{equation}

The Polya aspect of this approach is the following:

Truncate the Kernel $\Phi(t)$ of (2) and/or the integration range in (1) such that the resulting Fourier transform leads to a family of entire functions which only have real roots.

One such candidate is given in [C]: \begin{equation} \Phi_{\color{red}n}(t)=(1/2)\sum_{1\leqslant k\leqslant {\color{red}n}}\left(\phi_k(t)+\phi_k(-t)\right)=\Phi_{\color{red}n}(-t)\tag{4} \end{equation}

\begin{equation} \Xi_{\color{red}n}(z)=2\int_0^{(1/2)\log {\color{red}n}}\Phi_{\color{red}n}(t)\cos(zt)\mathrm{d}t=\Xi_{\color{red}n}(-z),\tag{5} \end{equation}

We refer to [D] and [E] for a near complete review on the zeros of entire functions as Fourier transforms.

The Hurwitz aspect of this approach is the following:

Corollary of Hurwitz's theorem in complex analysis (applied to our case) [F]:

If $\Xi(z)$ and $\{\Xi_n(z)\}$ are analytic functions on a domain $S_{1/2}(z)=\{z: 0<Im(z)<1/2\}$, $\{\Xi_n(z)\}$ converges to $\Xi(z)$ uniformly on compact subsets of $S_{1/2}(z)$, and all but finitely many $\Xi_n(z)$ have no zeros in $S_{1/2}(z)$, then either $\Xi(z)$ is identically zero or $\Xi(z)$ has no zeros in $S_{1/2}(z)$.

The functional equation for $\zeta(s)$ becomes $\Xi(-z)=\Xi(z)$. The candidate of $\Xi_n(z)$ in (5) automatically satisfies this functional equation.

Another benefit of Polya-Hurwitz approach is that the entrance barrier is relatively low (comparing to other approaches that usually require the advance knowledge of analytical number theory).

To get started, one only needs to know Fourier transform, basic complex analysis, some knowledge of entire functions, polynomials etc. So anyone who has math training with the college undergraduate math major may start to work on the Polya-Hurwitz approach and learn other necessary new math as he/she goes.

The most difficult part of Polya-Hurwitz approach seems to be the following:(for example,) proving that all the zeros of $\Xi_n(z)$ in (5) are real in $S_{1/2}(z)$.

One may need to have several iterations: guess one form of the Kernel like $\Phi_n(1,t)$ and complete the integration to get explicit expression for $\Xi_n(1,z)$. If all the zeros of $\Xi_n(1,z)$ are found not to be all real in $S_{1/2}(z)$, then move on to $\Phi_n(2,t)$ and $\Xi_n(2,z)$...

References:

[A] Titchmarsh,"The Theory of the Riemann Zeta-Function",
(1986).

[B] Edwards, "Riemann's Zeta Function", (1974).

[C] Shi, "On the zeros of Riemann Xi-function", (2017) arXiv:1706.08868.

[D] Dimitrov and Rusev, “ZEROS OF ENTIRE FOURIER TRANSFORMS” (2001), 108 page review paper.

[E] Hallum, “ZEROS OF ENTIRE FUNCTIONS REPRESENTED BY FOURIER TRANSFORMS” (2014), Master thesis.

[F] Conway, "Functions of One Complex Variable",(1978)

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