Questions tagged [riemann-zeta-function]

The Riemann zeta function is the function of one complex variable $s$ defined by the series $\zeta(s) = \sum_{n \geq 1} \frac{1}{n^s}$ when $\operatorname{Re}(s)>1$. It admits a meromorphic continuation to $\mathbb{C}$ with only a simple pole at $1$. This function satisfies a functional equation relating the values at $s$ and $1-s$. This is the most simple example of an $L$-function and a central object of number theory.

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Quasicrystals and the Riemann Hypothesis

Let $0 < k_1 < k_2 < k_3 < \cdots $ be all the zeros of the Riemann zeta function on the critical line: $$ \zeta(\frac{1}{2} + i k_j) = 0 $$ Let $f$ be the Fourier transform of the sum ...
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Geometric / physical / probabilistic interpretations of Riemann zeta($n>1$)?

What are some physical, geometric, or probabilistic interpretations of the values of the Riemann zeta function at the positive integers greater than one? I've found some examples: 1) In MO-Q111339 ...
50 votes
5 answers
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Motivated account of the prime number theorem and related topics

Though my own research interests (described below) are pretty far from analytic number theory, I have always wanted to understand the prime number theorem and related topics. In particular, I often ...
Sarah's user avatar
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49 votes
3 answers
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Is the Riemann zeta function surjective?

Is the Riemann zeta function surjective or does it miss one value?
Shimrod's user avatar
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49 votes
4 answers
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If the Riemann Hypothesis fails, must it fail infinitely often?

That is must there either be no non-trivial zeros off the critical line or infinitely many? I'm sure that no one believes otherwise, but I've never seen a theorem in the literature addressing this. ...
David Feldman's user avatar
45 votes
1 answer
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Is it possible to show that $\sum_{n=1}^{\infty} \frac{\mu(n)}{\sqrt{n}}$ diverges?

Let $\mu(n)$ denote the Mobius function with the well-known Dirichlet series representation $$ \frac{1}{\zeta(s)} = \sum_{n=1}^{\infty} \frac{\mu(n)}{n^{s}}. $$ Basic theorems about Dirichlet series ...
Jeremy Rouse's user avatar
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4 answers
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Why is so much work done on numerical verification of the Riemann Hypothesis?

I have noticed that there is a huge amount of work which has been done on numerically verifying the Riemann hypothesis for larger and larger non-trivial zeroes. I don't mean to ask a stupid question, ...
Hollis Williams's user avatar
43 votes
3 answers
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Could the Riemann zeta function be a solution for a known differential equation?

Riemann zeta function is a function of complex variable $s$ that analytically continous the sum of Dirichlet series .defined as :$$\zeta(s)=\sum_{n=1}^{\infty}\displaystyle \frac{1}{n^s} $$ for when ...
zeraoulia rafik's user avatar
43 votes
3 answers
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Is this integral representation of $\zeta(2n+1)$ known?

Background: I'm an undergraduate at an institution with no researchers in analytic number theory, and no ties to the analytic number theory community. I believe I have found what is, as far as I can ...
Andrew Knapp's user avatar
41 votes
6 answers
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"Long-standing conjectures in analysis ... often turn out to be false"

The title is a quote from a Jim Holt article entitled, "The Riemann zeta conjecture and the laughter of the primes" (p. 47).1 His example of a "long-standing conjecture" is the Riemann hypothesis,...
38 votes
2 answers
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Optimization problem arising from the study of zeta zeros

Motivation: The following problem arose in [1] while studying the vertical distribution of the zeros of the Riemann zeta-function. At the time, my collaborators and I were unable to solve it and I ...
Micah Milinovich's user avatar
37 votes
2 answers
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$\zeta(0)$ and the cotangent function

In preparing some practice problems for my complex analysis students, I stumbled across the following. It is not hard to show, using Liouville's theorem, that $$\pi\cot(\pi z)=\frac{1}{z}+\sum_{n=1}^\...
GH from MO's user avatar
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The Bourgain-Demeter-Guth breakthrough and the Riemann zeta function?

Yesterday Bourgain, Demeter and Guth released a preprint proving (up to endpoints) the so-called main conjecture of the Vinogradov's Mean Value Theorem for all degrees. This had previously been only ...
Mark Lewko's user avatar
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$\prod_{n=1}^{\infty} n^{\mu(n)}=\frac{1}{4 \pi ^2}$

When I tested this in Mathematica, I had expected it to say it did not converge. However, I got this: $$\prod_{n=1}^\infty n^{\mu(n)}=\frac{1}{4 \pi ^2}$$ Note: this is the reciprocal of (3) zeta-...
Fred Daniel Kline's user avatar
34 votes
7 answers
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Explicit formula for Riemann zeros counting function

I've often seen it stated (in vague terms) that there's a Fourier duality between the set of prime numbers and the set of nontrivial Riemann zeta zeros. Because there are various explicit formulae ...
user19727's user avatar
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2 answers
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Representations of $\zeta(3)$ as continued fractions involving cubic polynomials

$\zeta(3)$ has at least two well-known representations of the form $$\zeta(3)=\cfrac{k}{p(1) - \cfrac{1^6}{p(2)- \cfrac{2^6}{ p(3)- \cfrac{3^6}{p(4)-\ddots } }}},$$ where $k\in\mathbb Q$ and $p$ is a ...
Wolfgang's user avatar
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33 votes
7 answers
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Heuristic argument for the Riemann Hypothesis

Is there a heuristic argument that supports the validity of the Riemann hypothesis or are we just relying on numerical evidence? Moreover, what is the strongest theorem that supports the validity of ...
Mustafa Said's user avatar
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32 votes
2 answers
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Does the equation $1 + 2 + 3 + \dots = -\frac{1}{12}$ have a natural $p$-adic interpretation?

Consider the equation $$1 + 2 + 3 + 4 + \cdots = - \frac{1}{12},$$ "proved" by Ramanujan Euler. One correct way to interpret this is that $\zeta(-1) = - \frac{1}{12},$ where $\zeta(s) = \sum_{n = 1}^{\...
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Is there a closed form for $\int_0^\infty\frac{\tanh^3(x)}{x^2}dx$?

For $n\geqslant m>1$, the integral $$I_{n,m}:=\int\limits_0^\infty\dfrac{\tanh^n(x)}{x^m}dx$$ converges. If $m$ and $n$ are both even or both odd, we can use the residue theorem to easily evaluate ...
Wolfgang's user avatar
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What is Ricardo Pérez-Marco's eñe product? Does it explain his statistical results on differences of zeta zeros?

The number theory community here at University of Michigan is abuzz with talk of this paper recently posted to the arxiv. If you haven't seen it already, the punch line is that the global differences ...
Gene S. Kopp's user avatar
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A new way of approaching the pole of the Riemann zeta function - and a new conjectured formula

On the Wolfram page about the Euler-Mascheroni Constant $\gamma $, the following amazing limit is given without proof (referring to "personal communication"): $$\lim_{z\to\infty}\left[\zeta(\zeta(z))-...
Wolfgang's user avatar
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29 votes
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Good uses of Siegel zeros?

The short version of my question goes: What is known to follow from the existence of Siegel zeros? A longer version to give an idea of what I have in mind: The "exceptional zeros" of course first ...
Kálmán Kőszegi's user avatar
29 votes
1 answer
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Riemann's attempts to prove RH

I read somewhere that Riemann believed he could find a representation of the zeta function that would allow him to show that all the non-trivial zeros of the zeta function lie on the critical line. I ...
Mustafa Said's user avatar
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29 votes
3 answers
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$\zeta(n)$ as a mixed Tate motive

I am trying to understand why there exists, for each $n \geq 2$, a mixed Tate motive $M$ over $\mathbb{Q}$ such that $M \in Ext^1_{MT(\mathbb{Q})}(\mathbb{Q}(0), \mathbb{Q}(n))$ and $\zeta(n)$, ...
mtm93's user avatar
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1 answer
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The Riemann zeros and the heat equation

The Riemann xi function $\Xi(x)$ is defined, with $s=1/2+ix$, as $$ \Xi(x)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)=2\int_0^\infty \Phi(u)\cos(ux) \, du, $$ where $\Phi(u)$ is defined as $$ 2\sum_{...
Stopple's user avatar
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28 votes
2 answers
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What are some consequences of zero free strip of the Riemann zeta function?

A weaker version of the Riemann hypothesis is the claim that if $\zeta(s) = 0$ then $Re(s) \leq 1 - h$ for some constant $h> 0$. What would the consequences be of a result of this type?
Johnny T.'s user avatar
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Has Apéry's proof of the irrationality of $\zeta(3)$ ever been used to prove the irrationality of other constants?

Apéry's proof of the irrationality of $\zeta(3)$ astounded contemporary mathematicians for its wealth of new ideas and techniques in proving the irrationality of a known constant. It is often the case ...
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26 votes
5 answers
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Are the 'semi' trivial zeros of $\zeta(s) \pm \zeta(1-s)$ all on the critical line?

The proof that $\Gamma(z)\pm \Gamma(1-z)$ only has zeros for $z \in \mathbb{R}$ or $z= \frac12 +i \mathbb{R}$ has been given here: Are all zeros of $\Gamma(s) \pm \Gamma(1-s)$ on a line with real ...
Agno's user avatar
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26 votes
1 answer
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Why did Euler consider the zeta function?

Many zeta functions and L-functions which are generalizations of the Riemann zeta function play very important roles in modern mathematics (Kummer criterion, class number formula, Weil conjecture, BSD ...
Jojo's user avatar
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2 answers
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Are these two new ways of representing odd zeta values as integrals known?

This is inspired by the same beautiful integral expression for $\zeta(3)$ as this question, but goes in a slightly different direction. Writing the original integral in the form $$\int_0^1\frac{x(1-x)}...
Wolfgang's user avatar
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$P(s)=1-\sqrt{\frac{2}{\zeta(s)}-\sqrt{\frac{2}{\zeta(2s)}-\sqrt{\frac{2}{\zeta(4s)}-\sqrt{\frac{2}{\zeta(8s)}-...}}}}$

Vassilev-Missana - A note on prime zeta function and Riemann zeta function¹ claims the following remarkable identity: $$ P(s)=1-\sqrt{\frac{2}{\zeta(s)}-\sqrt{\frac{2}{\zeta(2s)}-\sqrt{\frac{2}{\zeta(...
Klangen's user avatar
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24 votes
1 answer
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Why these surprising proportionalities of integrals involving odd zeta values?

Inspired by the well known $$\int_0^1\frac{\ln(1-x)\ln x}x\mathrm dx=\zeta(3)$$ and the integral given here (writing $\zeta_r:=\zeta(r)$ for easier reading)$$\int_0^1\frac{\ln^3(1-x)\ln x}x\mathrm dx=...
Wolfgang's user avatar
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24 votes
1 answer
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How good is "almost all" when it comes to the Riemann Hypothesis?

Let $N(T)$ be the number of zeroes of the Riemann zeta function $\zeta$ having imaginary part strictly between $0$ and $T$, and let $N_0(T)$ be the number of those zeroes that also have real part ...
RHarris's user avatar
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23 votes
1 answer
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On an asymptotic formula of Keating and Snaith involving the Riemann zeta function

Keating and Snaith have a famous conjecture on the asymptotics of the integral $\int_0^T |\zeta(\frac 12+it)|^{2k}\, dt$, where $\zeta$ denotes the Riemann zeta function. See page 510 of the book ...
Richard Stanley's user avatar
23 votes
1 answer
3k views

More mysteries about the zeros of the Riemann zeta function

Update on 12/26/2020: I added the Appendix at the bottom: simplified formula for $|\zeta(s)|^2$, when $\frac{1}{2}<\Re(s)<1$. Update on 1/5/2020: I added the section "more interesting ...
Vincent Granville's user avatar
22 votes
2 answers
1k views

A closed form for an integral expressed as a finite series of $\zeta(2k+1)$, $\pi^m$ and a rational?

In this paper the following beautiful integral expression for $\zeta(3)$ is derived: $$\zeta(3)=\frac{1}{7}\,\int_0^{\pi} x\,(\pi-x)\csc(x)\, dx$$ In a comment at the end of this question, I ...
Agno's user avatar
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21 votes
2 answers
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What are the consequences of an ineffective proof of the Riemann Hypothesis?

Suppose a proof came out (and was verified by credible peer review) of the following statement: There is a $T_0$ such that for all $t>T_0$, all zeros $\zeta(\beta+it)=0$ have $\beta=1/2.$ where $...
Charles's user avatar
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21 votes
1 answer
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A multiple integral that seems related to the $\zeta$ function at even integers

I came across this integral that seems related to the Riemann zeta function $\zeta(2n)$ evaluated at even integers $2n \in 2\mathbb{Z}$. Letting $n$ be an even integer, define the multiple integral ...
Joe's user avatar
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21 votes
1 answer
680 views

On a pattern for upside-down Ramanujan pi formulas

Define, $$\lambda_n =\frac{(\tfrac12)_n}{(1)_n} =\frac{(\tfrac12)_n}{n!} =\frac{\tbinom{2n}{n}}{2^{2n}} =\binom{n-\tfrac12}{n}$$ with Pochhammer symbol $(x)_n$ and binomial $\tbinom{n}{k}$. I noticed ...
Tito Piezas III's user avatar
20 votes
4 answers
1k views

Bound on $L^2$ norm of $1/\zeta(1+i t)$?

What sort of bounds (explicit of preference) can one give for $$\int_T^{2 T} \frac{dt}{|\zeta(1+i t)|^2} \;\;\;\;\;?$$ Some obvious points: One can give a pointwise bound $\frac{1}{|\zeta(1+ it)|} \...
H A Helfgott's user avatar
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20 votes
3 answers
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Jensen Polynomials for the Riemann Zeta Function

In the paper by Griffin, Ono, Rolen and Zagier which appeared on the arXiv today, (Update: published now in PNAS) the abstract includes In the case of the Riemann zeta function, this proves the ...
Stopple's user avatar
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20 votes
2 answers
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Zeros of higher derivatives of $\zeta(s)$

Zeros of successive higher order derivatives of the Riemann zeta function seem to cluster along roughly horizontal lines. Is there a heuristic explanation of why this happens (especially inside the ...
Stopple's user avatar
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20 votes
1 answer
701 views

On the equation $\zeta(s) = F(s)+F(s+1)$

Define the function $F(s)$ as the Dirichlet series $$ F(s) = \sum_{n=1}^\infty \frac{1}{(n+1)n^{s-1}}, $$ which converges for $\operatorname{Re}(s)>1$. Has anyone seen/studied this function before? ...
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19 votes
3 answers
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Are the nontrivial zeros of the Riemann zeta simple?

A few years ago, I found on arXiv an article in which the authors (I think they were at least two to write it) claimed to have proven that the non trivial zeros of the Riemann zeta function were all ...
Sylvain JULIEN's user avatar
19 votes
4 answers
2k views

What are the obstructions to showing that $\zeta$ doesn't vanish on the strip $1- \varepsilon < {\rm Re}(s) \leq 1$

Most (if not all) of the proofs of the Prime Number Theorem that I have seen in the literature rely on the fact that the Riemann zeta function, $\zeta(s)$, does not vanish on the line ${\rm Re}(s) = 1$...
Mustafa Said's user avatar
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19 votes
4 answers
3k views

zeta(3) in terms of derivatives of zeta at 1/2 and pi

Got numerical support that for odd $n$, $\zeta(n)$ might be expressed in terms of the derivatives of $\zeta(\frac12)$. Based on More Zeta Functions for the Riemann Zeros, Andre Voros, p.12, Table 3: ...
joro's user avatar
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18 votes
2 answers
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How did Riemann calculate the first few non-trivial zeros of the zeta-function?

Does anyone know how Riemann calculated the first few non-trivial zeros of the Zeta function? I am wondering if he approximated the integral, $\frac{1}{2 \pi i} \int_{R} \frac{{\xi}^\prime(z)}{\xi (z)...
Mustafa Said's user avatar
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18 votes
1 answer
674 views

Could computing the next prime in a finite Euler product be made rigorous?

It is well known that: $$\zeta(s):=\prod_{n=1}^{\infty} \frac{1}{1-p_n^{-s}} \qquad \Re(s) \gt 1$$ with $p_n =$ the $n$-th prime. It also known that: $$\zeta(2n):= \frac{(-1)^{n+1} B_{2n}(2\pi)^{2n}}{...
Agno's user avatar
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18 votes
1 answer
1k views

stable homotopy groups and zeta function

I have heard during a discussion that there is a well known relation between the stable homotopy groups of a sphere (more precisely the order of stable homotopy groups of localized sphere spectrum ...
Max's user avatar
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18 votes
1 answer
1k views

Riemann's $\zeta$ function and the uniform distribution on $[-1,0]$

https://math.stackexchange.com/questions/64566/riemanns-zeta-function-and-the-uniform-distribution-on-1-0 Stackexchange isn't getting really excited about this, so here it is. The $n$th cumulant of ...
Michael Hardy's user avatar

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