**12**

votes

**2**answers

574 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 ...

**1**

vote

**1**answer

560 views

### Sharpening a bound on $\zeta'(s)$

I want to find an upper bound for $\zeta'(s)$ along a vertical line $\Re(s)=b$, where $-1<b<0$.
One way to do this is using $$\frac{\zeta'(b+iT)}{\zeta(b+iT)}=O_b(\log T)$$ and ...

**1**

vote

**0**answers

116 views

### Simultaneous vanishing of convolutions of Mertens function with itself

By Landau's theorem on Dirichlet series, we know that all the step functions ($k\geq 1$)
$$M_k(x)=\frac{1}{2\pi i}\int^{2+i\infty}_{2-i\infty}\frac{x^sds}{\zeta^k(s)s}=\sum_{n\leq ...

**28**

votes

**1**answer

1k views

### 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 ...

**7**

votes

**2**answers

364 views

### Visibility interpretation of Riemann zeta zeros on the critical line?

This is a long shot, but ...
The fraction of $\mathbb{Z}^2$ lattice points
visible from the origin
$1/\zeta(2)=6/\pi^2 \approx 61$%.
The fraction of $\mathbb{Z}^3$ lattice points visible
from the ...

**8**

votes

**0**answers

851 views

### The zeta function and classical mechanics

In this paper, Guilherme França and André LeClair show that $$\gamma_{y}\sim 2 \pi \left(y-11/8\right)/W\left((y-11/8)e^{-1}\right)$$ where $W$ is the Lambert W function, and $\gamma_{y}$ is the ...

**5**

votes

**1**answer

334 views

### Why do $12$ and $120$ occur very often in the denominators of $\zeta(-n)$ for odd $n$?

$\zeta(-n) = - \dfrac{B_{n+1}}{n+1}$
$\zeta(-2n) = 0$
$\zeta(-1) = - \dfrac{1}{12}$
$\zeta(-3) = \dfrac{1}{120}$
$\zeta(-5) = - \dfrac{1}{252}$
$\zeta(-7) = \dfrac{1}{240}$
$\zeta(-9) = - ...

**3**

votes

**1**answer

425 views

### what would be the consequences on the distribution of primes of $\Lambda=\infty$?

It is widely believed that the quantity $\Lambda:=\lim\sup\dfrac{t_{n+1}-t_{n}}{2\pi/\log t_{n}}$, where $t_{n}$ is the imaginary part of the $n$-th non-trivial zero on the critical line of the ...

**0**

votes

**0**answers

81 views

### Conjectured alternate form for vanishing of $\Re\zeta(1/2+it)$ except at zeros

Heavily based on Agno's question.
Define:
$$ \chi(s)=\pi^{-(\frac{s}{2})} \Gamma(\frac{s}{2}) $$
Agno conjectured: only for $\sigma=\frac12$, $\Re(\chi(s)) = \Re(\zeta(s)) =0$ is always true, ...

**1**

vote

**0**answers

114 views

### Except for a finite few outside the strip, do all complex zeros of $\zeta(a+s)\pm \zeta(a+1-s)$ reside on the critical line for all $a\lt 0$?

Assume $a \in \mathbb{R}$ and $s \in \mathbb{C}$.
Numerical evidence suggests that all complex zeros, except for a finite few outside the strip, of:
$$\zeta(a+s)\pm \zeta(a+1-s)$$
lie on the line ...

**2**

votes

**1**answer

255 views

### Is there anything known about the complex zeros of this integral related to $\zeta(s)$?

The right-hand side of the well known equation:
$$\displaystyle \pi^\frac{-s}{2}\,\Gamma\left(\frac{s}{2}\right)\, \zeta(s) =\int_1^{\infty} \left({x}^{\frac{s}{2}} + ...

**6**

votes

**0**answers

182 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 ...

**3**

votes

**1**answer

243 views

### expressing $\log(\left \lfloor x \right \rfloor!)$ in terms of zeta-zeros

referring to a question i posted on MS, I post it here, as I didn't get an answer:
let $\psi(x)$ be the second Chebyshev Function. By the definition of this summatory function, and the fundamental ...

**0**

votes

**0**answers

50 views

### $\eta(s)$ expressed as an 'alternating' sum of Hurwitz Zetas. Why does it only work for sums with an even number of terms?

It is known that:
$$\zeta(s)= a^{-s}\,\sum_{k=1}^{a} \zeta_H\left(s,\frac{k}{a}\right)$$
is valid for all $a \in \mathbb{N}$ and all $s \in \mathbb{C}\,/1$, with $\zeta_H$ being the Hurwitz zeta ...

**2**

votes

**2**answers

317 views

### Conjectured relation between alternating Prime zeta series and Riemann zeta

Let $P(s)$ be the Prime zeta function.
Numerical evidence suggests these identities:
$$ \sum_{k=1}^\infty \frac{(-1)^{k}P(3k)}{k}=\log{\bigg(\frac{1}{945}\frac{\pi^6}{\zeta(3)}\bigg)}\qquad\quad ...

**0**

votes

**1**answer

126 views

### Estimating the height required to find a given small value of $|\zeta(s)|$ near the line $\sigma=1$

There are some qualitative theorems of Bohr, Jessen and Titchmarsh (e.g. The Theory of the Riemann zeta function, E.C. Titchmarsh, pages 306-308) proving that there is a $K=K(a,\alpha,\beta)$ such ...

**3**

votes

**1**answer

114 views

### On the domain of convergence of usual definition of Riemann zeta function [closed]

If we define Riemann zeta function as it is usually defined so that $$\zeta (s)=\sum_{n=1}^{\infty}n^{-s}=\sum_{n=1}^{\infty}e^{-s\ln n}$$ then we can rewrite it as:
$$\zeta ...

**3**

votes

**2**answers

288 views

### Are there any new results on approximating Riemann $\Xi$ function by Polya-like Fourier transforms?

I posted [this question][1] at math.stackexchange.com and was told that it is more appropriate to post this research related question here at mathoverflow.
So I re-post it below.
Riemann $\Xi(z)$ ...

**5**

votes

**3**answers

390 views

### Bounds on horizontal minima of the Riemann zeta function

It is known that $\zeta(s)$ has an infinity of zeros in the strip $0<\sigma<1$ and that those zeros become closer together as $t\rightarrow\infty$. More precisely, Littlewood showed that there ...

**1**

vote

**0**answers

80 views

### Are all complex zeros of $Li_s(i)\, + \, Li_{1-\overline{s}}\,(-i)$ equal to the $\rho$'s?

Take the well known square relationship for polylogarithms:
$$Li_s(z)\, + \, Li_{s}(-z)=2^{1-s}Li_s(z^2)$$
Assume $z=i$:
$$Li_s(i)\, + \, Li_{s}(-i)=2^{1-s}Li_s(-1)=-2^{1-s}\,\eta(s)$$
with ...

**1**

vote

**0**answers

71 views

### Are all complex zeros of $Li_s(z)\, \pm \, Li_{1-s}(z)$ on the critical line or outside the critical strip for $z \le -1$?

This question loosely builds on this one, however is a bit simpler and I found the results to be more robust.
It seems that all zeros in the critical strip $0 \lt \Re(s) < 1$ of:
$$Li_s(z)\, \pm ...

**24**

votes

**3**answers

3k views

### $\prod_{n=1}^{\infty} n^{\mu(n)}=\frac{1}{4 \pi ^2}$

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) ...

**32**

votes

**6**answers

3k views

### 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 ...

**1**

vote

**0**answers

144 views

### Contour plots of Riemann zeta-function

A glimpse of figures in this preprint seems to suggest that curves $\Re{\zeta(s)}=0$ (or $\Im{\zeta(s)}=0$) do not touch each other in the half-plane $\Re{s}>1$.
Question: Is there any ...

**6**

votes

**0**answers

219 views

### Are all complex zeros of $\frac{\Gamma(s)}{z}Li_s(z) \, \pm \, \frac{\Gamma(1-s)}{z}Li_{1-s}(z)$ on the critical line for all $z \lt 1$?

Numerical evidence suggests that all complex zeros residing in the critical strip $0 < \Re(s) < 1$ of:
$$\frac{\Gamma(s)}{z}Li_s(z) \, \pm \, \frac{\Gamma(1-s)}{z}Li_{1-s}(z)$$
are on the ...

**3**

votes

**1**answer

376 views

### What is the (fractional) half-derivative of zeta at $s=0$ (and how to compute it)?

(I asked this in MSE before but there was only a general reference which did not help for my specific question)
I think I understood the concept of fractional derivatives applied to ...

**3**

votes

**1**answer

248 views

### Questions about the Riemann Zeta Function

How many contiguous zeros of zeta are known, to what height
How many contiguous primes are known, to what height
How many zeta zeros determine how many primes, to what exactness
For example, would ...

**0**

votes

**0**answers

198 views

### Complex zeros of $\zeta'(s)/\zeta(s) + \zeta'(1-s)/\zeta(1-s) $ = simpler expression (except at zeta zeros)

Let $ G(s) := \frac{\zeta'(s)}{\zeta(s)} + \frac{\zeta'(1-s)}{\zeta(1-s)}$
where $s$ is not a zero of zeta.
$G$ has real zeros and a pair of complex zeros near $\frac12 \pm 6i$.
Partial results:
By ...

**4**

votes

**0**answers

55 views

### Szegő curve for partial sum of Taylor series of Riemann $\Xi(z)$ function

I am sorry that this is long post. But it might be of interest to you.
This post is related to zeros of partial sum of Taylor series of $e^x-1$.
Entire functions $e^z$, $\cos(z)$, and $\sin(z)$ can ...

**1**

vote

**2**answers

189 views

### zeta-function regularized integrals

I gather that the following two identities about $\xi(3)$ hold via some notion of zeta-function regularized integrals.
$\xi(3) = \frac{(2\pi)^3}{3}\int _0 ^\infty d\lambda \frac{\sqrt{\lambda} }{1 + ...

**2**

votes

**1**answer

119 views

### A Hadamard product of the zeros of the Riemann integral. Does it put any constraints on where the $\rho$'s can reside in the critical strip?

I have deleted a previous, now obsolete question on the same topic.
Take the well-known Riemann integral:
$$\displaystyle \pi^{-\frac{s}{2}}\,\Gamma\left(\frac{s}{2}\right)\, \zeta(s) ...

**25**

votes

**1**answer

810 views

### 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 ...

**7**

votes

**1**answer

547 views

### Continued fraction representation of Zeta

A question at math.SE is asking for references. The fraction is quite nice! Check it out and post some references if you know of any.
I found this at arxiv, but it doesn't apply to Zeta.

**0**

votes

**0**answers

121 views

### Does Riemann's explicit formula imply invariance of the prime gaps distribution under a Fourier-like transform?

Loosely speaking, Riemann's explicit formula states that there exists a Fourier-type duality between the primes and the non trivial zeroes of the Riemann zeta function. Does this mean that the ...

**1**

vote

**0**answers

72 views

### Do we know a lower bound for the number of critical zeros of the Riemann zeta-function with irrational imaginary part?

If I'm not mistaken, the imaginary parts of the critical zeros of the Riemann Zeta function are conjectured to be linearly independent over $\mathbb{Q}$, but I think we're very far from proving such a ...

**3**

votes

**1**answer

254 views

### leading-order behaviour of riemann zeta function?

Is there any 'guess' as to how the Riemann zeta function $\zeta(\sigma+it)$ (or its modulus) behaves to leading order as $t\rightarrow\infty$, for fixed $\sigma$ in the critical strip? Obviously this ...

**3**

votes

**1**answer

303 views

### Verifying very high Riemann zeros.

Using some newly derived formulas for the n-th Riemann zero on the critical line,
I calculated the 10^(10^6)'th zero to 1 million decimal places rather easily.
Can anyone suggest an alternative way to ...

**8**

votes

**1**answer

194 views

### Sharpest bound on the zero free region of $\zeta^{\prime}$?

I'm interested in calculating all of the zeroes of the first derivative of the Riemann $\zeta$ function up to an arbitrary height. I know that (on the RH), all of these zeroes will have real part $\ge ...

**4**

votes

**0**answers

86 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 ...

**19**

votes

**6**answers

3k views

### 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 ...

**23**

votes

**1**answer

928 views

### 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 ...

**0**

votes

**1**answer

168 views

### A couple of facts on the non-trivial zeros of the Riemann Zeta function

This question might be more suitable for http://math.stackexchange.com. I'm not sure about the differences between that website and this website (http://mathoverflow.net), so I'll try it here first.
...

**1**

vote

**0**answers

72 views

### Question about the zeros of the sum/difference of two finite Euler products

The conjecture Are all zeros of $\zeta(0+s) \pm \zeta(0-s)$ except a finite few on the line $\Re(s)=0$? was shown to be unconditionally true.
The proof can even be extended towards the domain ...

**7**

votes

**3**answers

851 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 ...

**6**

votes

**1**answer

457 views

### Are all complex zeros of $\zeta(s) \pm \zeta(-s)$ on the line with $\Re(s)=0$?

My conjecture is that all zeros in the strip $-1 \le \Re(s) \le 1$ of $\zeta(s) \pm \zeta(-s)$ are on the line $\Re(s)=0$.
I did find three complex zeros for $\pm =+$ (i.e. 12 in total) and two ...

**3**

votes

**1**answer

192 views

### reference for Lindelof 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 Lindelof Hypothesis is true (that is $\zeta(\sigma+it)=O(t^\epsilon)$ as $t\rightarrow\infty$), then ...

**2**

votes

**2**answers

325 views

### Riemann Z function, bounds on number of non-trivial zeros along horizontal lines, rather than vertical ones

Concerning the non-trivial zeros of the Riemann Zeta function, one can find quite a lot of literature on:
the rate of growth of the number of zeros along the vertical critical line,
the zero-free ...

**2**

votes

**0**answers

334 views

### The simple zero conjecture for the Riemann zeta function

The simple zero conjecture says that all zeros of the Riemann zeta function are simple.
Suppose the conjecture is not true. Namely there is an $s$ in the complex plane such that $\zeta(s)=0$ and the ...

**3**

votes

**1**answer

303 views

### Series of the inverse quadratic trinomial

Maybe it's a very simple question, but I have a problem with the following series
$$\sum\limits_{n=1}^{\infty}\frac{1}{n^2+pn+q},$$
where $p, q \in \mathbb{R}$. I know about five ways how to calculate ...

**42**

votes

**5**answers

4k views

### 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 ...