Skip to main content

All Questions

Filter by
Sorted by
Tagged with
52 votes
3 answers
6k views

Is the Riemann zeta function surjective?

Is the Riemann zeta function surjective or does it miss one value?
Shimrod's user avatar
  • 2,375
49 votes
4 answers
6k views

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
37 votes
2 answers
3k views

$\zeta(0)$ 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
  • 105k
35 votes
7 answers
6k views

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
  • 3,699
29 votes
1 answer
2k views

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
  • 3,699
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
20 votes
1 answer
745 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? ...
Dan Romik's user avatar
  • 2,549
18 votes
2 answers
5k views

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
  • 3,699
17 votes
2 answers
2k views

Algebraic independence of shifts of the Riemann zeta function

Let $\zeta(s)$ denote the Riemann zeta function. Is the set $\{ \zeta(s-j)\, \colon\, j\in\mathbb{Z}\}$, or even $\{\zeta(s-z)\, \colon\, z\in\mathbb{C}\}$, algebraically independent over $\mathbb{C}$?...
Richard Stanley's user avatar
16 votes
1 answer
2k views

Certain functional equations for the Riemann Zeta function?

Referring to this question I asked on math.SE. I am posting a more generalized question here, for answers and further inquiry. For the Riemann zeta function, we know of the standard functional ...
Roupam Ghosh's user avatar
16 votes
2 answers
2k views

On the Universality of the Riemann zeta-function

Hi, I have a question regarding the universality property of the Riemann zeta-function. I am no expert on this, so I'd be glad for any relevant reference. First, recall Voronin's remarkable theorem ...
15 votes
5 answers
2k views

Zeros of the derivative of Riemann's $\xi$-function

The Riemann xi function $\xi(s)$ is defined as $$ \xi(s)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s). $$ It is an entire function whose zeros are precisely those of $\zeta(s)$. Since $\xi$ is real ...
Stopple's user avatar
  • 11.1k
14 votes
2 answers
2k views

Is $\pi (x)=\operatorname{R}(x)-\sum_{\rho}\operatorname{R}(x^{\rho})$ correct at all?

It should be the case that, in some appropriate sense $$\pi (x)\sim \operatorname{Ri}(x)-\sum_{\rho}\operatorname{Ri}(x^{\rho}) \tag{4}\label{386213_4}$$ with $\operatorname{Ri}$ denoting the Riemann ...
Wane's user avatar
  • 83
14 votes
1 answer
1k views

What is the analytic continuation of $\varphi(s)=\sum_{n \ge 1} e^{-n^s}?$

My research has lead me to the following function that I'm trying to continue. 3 Months ago I posted this question to MSE, and have placed 3 bounties on the question, but haven't received an answer, ...
geocalc33's user avatar
  • 105
14 votes
1 answer
1k views

On meromorphic continuation of zeta function(s) and special values at negative integers

Euler developped (at least) two different approaches in order to calculate the values $\zeta(-m)$ of the zeta function $$\zeta(s) = \sum_{n\geq 1} \frac{1}{n^s}$$ at non-positive integers. In one ...
user5831's user avatar
  • 2,029
14 votes
1 answer
1k views

The location of the zeros of the "new" function $\Psi(s)=\sum _{n=1}^{\infty }{\frac{1}{n!^s}}$

Defining the Psi function as $$\Psi(s)=\sum _{n=1}^{\infty }{\frac{1}{n!^s}}$$ and by studying, from a numerical point of view, the location of the zeros in the complex plane up to $0<\Im(s)<...
Roberto Trocchi's user avatar
13 votes
3 answers
1k views

Is anything known about the series $\sum_{n=0}^{\infty} x^{\sqrt{n}} $?

It's well known that there are a shocking number of identities for the usual Jacobi theta function $$ \theta_3(x) = \sum_{n=-\infty}^{\infty} x^{n^2}. $$ So I wanted to turn my attention to slowly ...
Sidharth Ghoshal's user avatar
13 votes
2 answers
725 views

Special values of $\zeta$ outside the real line and the critical strip

The values of Riemann's function at the integers have been extensively studied. I was wondering, is there anything interesting known (or conjectured) to happen arithmetically outside the real line (...
Myshkin's user avatar
  • 17.6k
12 votes
2 answers
555 views

$\zeta^{(k)}(s) < 0$ for $s\in (0,1)$

A bit of plotting suggests that $\zeta^{(k)}(s) < 0$ for all $s\in (0,1)$ and all integers $k\geq 0$. (Or, what is the same: $\zeta^{(k)}(s)$ has no zeroes on $(0,1)$.) Is there a brief, clean ...
H A Helfgott's user avatar
  • 20.1k
12 votes
4 answers
916 views

non-trivial zeros of partial zeta functions

Let $N,a\in\mathbf{Z}_{\geq 1}$. Define a partial $\zeta$-function as $$ \zeta(s;N,a):=\sum_{\substack{n\geq 1\newline n\equiv a\pmod{N}}} \frac{1}{n^s} $$ where $Re(s)>1$. Let $\omega$ be either ...
Hugo Chapdelaine's user avatar
12 votes
2 answers
1k views

Has there been further work on Bender-Brody-Müller approach to RH?

Earlier this year (April 4, 2017), a seemingly tantalizing approach of the Riemann Hypothesis based on ideas dating back to Hilbert and Pólya by Bender, Brody and Müller was made publicly available. I ...
Sylvain JULIEN's user avatar
11 votes
0 answers
530 views

Cesaro summation of a particular Dirichlet series associated with $\zeta(s)$

If you've investigated the error in Perron's formula in general, you've probably noticed that Cesaro summation $$\lim_{x\rightarrow\infty}\sum_{n\leq x} \left(1-\frac{\log n}{\log x}\right)\frac{a_n}{...
Kevin Smith's user avatar
  • 2,480
10 votes
3 answers
853 views

The smallest nontrivial zero of the Riemann zeta function

Consider the Riemann zeta function $$\zeta(s)=\dfrac{1}{1-2^{1-s}}\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^s},\operatorname {R}(s)>0.$$ Riemann suggested that all nontrivial zeroes lie on the line $\...
わくわく's user avatar
9 votes
2 answers
1k views

On the error term of the Riemann explicit formula

Let: $\rho$ be a non-trivial zero of the Riemann zeta function, $\Lambda$ be the von-Mangoldt function and $\psi(x) =\sum_{n \leq x} \Lambda(n)$. What is the best known upper bound for $$f(x, T) := \...
Q_p's user avatar
  • 1,019
9 votes
1 answer
853 views

Moments of the Riemann zeta function

Is it possible to get an upper bound better than $\ll_\sigma T^{3/2-\sigma}$ for $$\int_{0}^{T}|\zeta (\sigma +it)|\,dt,\qquad 0<\sigma<1/2\,?$$
user155294's user avatar
9 votes
1 answer
741 views

Three questions about three functions similar to $\sin,\cos$

In The Basel problem revisited? a question about the function, similar to sinc, $f(x)$ was asked: $$f(x) = \prod_{n=1}^\infty \left ( 1+ \frac{x^3}{n^3} \right ) = \prod_{n=1}^\infty \left ( 1+ \frac{...
mathoverflowUser's user avatar
9 votes
1 answer
939 views

A question on the Riemann zeta function

Yesterday, a certain very talented and passionate young student from Southern Africa asked me the following question about the Riemann zeta function $\zeta(s)$. He says he "thinks" he knows the answer,...
non-number theorist's user avatar
9 votes
2 answers
2k views

References on Taylor series expansion of Riemann xi function

I am looking for the references on Taylor series expansion of Riemann xi function at $\frac{1}{2}$. $$ \xi (s)=\sum_0^{\infty}a_{2n}(s-\frac{1}{2})^{2n}$$ where $$a_{2n}=4\int_1^{\infty}\frac{d[x^{3/...
mike's user avatar
  • 603
9 votes
0 answers
414 views

From holes in the image of peculiar functions to new perspective on the Riemann Hypothesis

I am working with the Dirichlet eta function $\eta(z)$, with $z=\sigma+it$, $\sigma > \frac{1}{2}$, and $t>0$. Let us define $$\eta_n(z,\gamma)= \sum_{k=1}^n (-1)^{k+1}\lambda_k^{-\sigma} e^{-it\...
Vincent Granville's user avatar
9 votes
0 answers
264 views

Functional equation or analytic continuation of certain approximations to $\zeta^z(s)$?

Let $z$ be a complex number and $\omega(n)$ denote the number of distinct prime factors of the natural number $n$. I am considering the arithmetic functions $|\mu(n)|z^{\omega(n)}$ and their ...
Kevin Smith's user avatar
  • 2,480
8 votes
3 answers
1k views

Objections to and arguments for the simplicity of all Riemann zeros

It seems to be that the simplicity of all the zeros is quite widely accepted as a working hypotheses, and it is known that a positive proportion are as such. Titchmarsh explains in the last chapter ...
Kevin Smith's user avatar
  • 2,480
8 votes
2 answers
296 views

How to formalize the *loci of equal arg($\zeta(s)$)* ("isogones") in the near of a nontrivial root

(This is an extension and specification of a question which I initially asked in MSE having now one comment (which I could not yet digest completely) and which I also detailed further (after working ...
Gottfried Helms's user avatar
8 votes
2 answers
2k views

Pair correlation for the Riemann zeros and $(\zeta^\prime(s)/\zeta(s))^\prime$

Added Background: The pair correlation of the zeros of the Riemann zeta function is influenced by the the derivative of the logarithmic derivative $(\zeta^\prime(s)/\zeta(s))^\prime$; see for example ...
Stopple's user avatar
  • 11.1k
7 votes
2 answers
788 views

Reference request for the explicit formula for $\sum_{n\leq x} \Lambda(n)n^{-s}$

Denote by $\Lambda(n)$ the von Mangoldt function, which is equal to $\log p$ if $p\geq 2$ is a prime, and $0$ otherwise. Let $\rho$ denote a complex zero of the Riemann $\zeta$-function. If I recall ...
Q_p's user avatar
  • 1,019
7 votes
2 answers
719 views

On the asymptotic behaviour of the series $\sum_{n=1}^{\infty} \left ( \frac{\zeta(ns)}{n}+\frac{s}{1-ns}\right )$ near $s=0$

I am interested in determining the behaviour of the the series/function $$f(s)=\sum_{n=1}^{\infty} \left ( \frac{\zeta(ns)}{n}+\frac{s}{1-ns}\right )$$ near $s=0$. It is clear that $f(0)$ is undefined....
Tian Vlašić's user avatar
6 votes
1 answer
900 views

What is the best known upper bound for $\frac{1}{\zeta'(\rho)}$ assuming the SZC but not the RH?

Let $\zeta$ denote the Riemann zeta function and let $\rho$ denote one of its complex zeros. What is the best known upper bound for $\frac{1}{\zeta'(\rho)}$ assuming that all zeros are simple (SZC), ...
user257465's user avatar
6 votes
1 answer
4k views

About the logarithmic derivative of the Riemann zeta function

Let $\rho=\beta+i\gamma$ a non-trivial zeros of the Riemann zeta function and $s=\sigma+it$ a complex number. It is possible to prove that $$\frac{\zeta'}{\zeta}\left(s\right)=\sum_{\left|t-\gamma\...
User's user avatar
  • 219
6 votes
2 answers
784 views

On some analytic property of the Riemann zeta function

Denote by $\zeta$ the Riemann zeta function. For $\Re(s)=\sigma>0$, it is well known that $$\sum_{n\leq x} n^{-s} = \zeta(s) + \frac{x^{1-s}}{1-s}+ O(x^{-\sigma}).$$ But do there exist infinitely ...
user avatar
6 votes
1 answer
247 views

Convergence and meromorphic continuation of a Dirichlet series under RH

Consider a Dirichlet series $F(s) = \sum_{n\geq1} \frac{a_{n}}{n^{s}}$ such that $a_{n} = O(1)$. Suppose the Dirichlet series $$\zeta(s)F(s)=\sum_{n\geq1} \frac{(a\star1)(n)}{n^{s}}$$ converges ...
 Babar's user avatar
  • 611
6 votes
1 answer
279 views

Can Voronin's universality theorem be used to show that $\sigma\circ\zeta=\zeta\circ\sigma$ implies $\sigma$ continuous?

Let $\sigma$ be a field automorphism of $\mathbb{C}$ that commutes with the Riemann Zeta function. Can we use Voronin's universality theorem to prove that $\sigma$ is necessarily continuous? Thanks in ...
Sylvain JULIEN's user avatar
6 votes
1 answer
383 views

Tight error terms for partial sums $\sum_{n\leq x} 1/n^s$

(a) Let $s>1$, $x>0$ be real. Then it is not hard to see that $$\sum_{n\leq x} \frac{1}{n^s} \leq \zeta(s) - \frac{1}{(s-1) x^{s-1}} + \frac{1}{2 x^s},$$ basically because $x\mapsto 1/x^s$ is ...
H A Helfgott's user avatar
  • 20.1k
5 votes
1 answer
765 views

Reference for Lindelöf Hypothesis implying finitely many zeros off critical line?

Can anyone give me a reference for the following theorem on the Riemann zeta function? If the Lindelöf Hypothesis is true (that is $\zeta(\sigma+it)=O(t^\epsilon)$ as $t\rightarrow\infty$), then ...
Harry Macpherson's user avatar
5 votes
1 answer
426 views

Lindelöf hypotheses for derivatives of zeta

The Lindelöf hypothesis says that if we have: $$\zeta(\sigma+iT)=\mathcal O(T^a)$$ Then if one considers $\sigma=1/2$ then $\inf a=0$. Further, from convexity and the functional equation this implies ...
psubodiosa's user avatar
5 votes
2 answers
850 views

Local phase statistics of the nontrivial Riemann zeros

(The question is inspired by Owen Maresh's post) The local phase of a nontrivial zero $s$ of the Riemann $\zeta$ is the argument of $\zeta'(s)$. Numerical results on the first 10000 zeros suggest ...
LeechLattice's user avatar
  • 9,501
5 votes
1 answer
291 views

Asymptotics of the Liouville sum at the primes

Let $\lambda$ denote the Liouville function, and let $L(x):=\sum_{n \leq x} \lambda(n)$ be the Liouville sum. Define $c$ to be the supremum of the real parts of the zeros of the Riemann zeta function. ...
user501735's user avatar
5 votes
1 answer
660 views

Maximal analytic continuation of $\varphi(s)=\sum_{n\ge1} e^{-n^s}$

About 6 months ago I asked for an analytic continuation of $\varphi(s)=\sum_{n\ge1} e^{-n^s}.$ What's the maximal analytic continuation of $\varphi(s)?$ Doing this will help me better understand how ...
geocalc33's user avatar
  • 105
5 votes
1 answer
654 views

Xi Function on Critical Strip - Mellin Transform

Story I'm trying to prove following identity $$\int_0^\infty \frac{\Xi(t)}{t^2 + \frac{1}{4}} \cos(xt) dt = \frac{1}{2} \pi (e^{\frac{1}{2}x} - 2e^{-\frac{1}{2}x} \psi(e^{-2x}))$$ where $$\psi(x)=\...
fje's user avatar
  • 183
5 votes
0 answers
321 views

Approximating $\zeta^{(r)}(s)$ by a sum

Let $\eta:[0,\infty)\to [0,\infty)$ be compactly supported, continuous and piecewise $C^1$, with its derivative $\eta'$ being of bounded variation. It is completely unsurprising that one can prove (...
H A Helfgott's user avatar
  • 20.1k
5 votes
0 answers
260 views

What is the winding behavior of the Riemann zeta function around zero along the line $s=1+it$?

Let $\phi: \mathbb R \setminus \{0\} \to S^1 \subset \mathbb C$ be defined by $$\phi(t)= \zeta(1+it)/|\zeta(1+it)|$$ (the nonvavishing of the denominator being a bit weaker than the prime number ...
Tim Campion's user avatar
  • 63.9k
5 votes
0 answers
161 views

On the asymptotics of some sum involving the Mertens function

Let $a_n$ be a sequence of nonnegative real numbers such that $\sum_{n\leq x} a_n \gg \frac{\sqrt x}{\log x}$ for large enough $x$. Denote by $\mu$ the Mobius function, and let $M(N)=\sum_{n\leq N} \...
Q_p's user avatar
  • 1,019