**4**

votes

**1**answer

208 views

### Do we know an upper bound for the multiplicity of the non-trivial zeros of Zeta?

In Are the non trivial zeros of Zeta simple?, I asked whether it was known that all non-trivial zeros of the Riemann Zeta function were simple or not. It appears that such a proof is missing. But are ...

**5**

votes

**1**answer

163 views

### Conjecture of Spira on the zeros of $\zeta^\prime(s)$

Let $N(T)$ be the number of complex zeros of $\zeta(s)$ with imaginary part between $0$ and $T$, and let $N_k(T)$ be the analogous counting function for the $k$th derivative $\zeta^{(k)}(s)$. Based ...

**1**

vote

**0**answers

198 views

### Are these valid expansions of the Riemann $\xi(s)$ function in the Hadamard product?

In this post I derived for $s=a + ti$, that assuming the RH, the following should be true:
$$\displaystyle \frac{\xi(\frac12 - a + s)}{\xi(\frac12 - a)} = \prod_{n=1}^\infty \left(1- \frac{s}{\mu_n} ...

**3**

votes

**0**answers

202 views

### Does the difference of two converging infinite series correctly induce the non-trivial zeros of $\zeta(s)$?

The following analytic continuation for $\zeta(s)$ towards $\Re(s)>-1$ derived here:
$$\displaystyle \zeta(s) = \frac{1}{2\,(s-1)} \left(s+1+ \sum _{n=1}^{\infty } \left( {\frac ...

**10**

votes

**2**answers

708 views

### How did Riemann calculate the first few non-trivial zeros of Zeta?

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 ...

**3**

votes

**1**answer

311 views

### On link between Riemann hypothesis and partial GRH

Is there a way to show that if the Riemann hypothesis holds for Dirichlet L-function associated to primitive Dirichlet character (excluding trivial character $\chi(1)$ which could be qualified of ...

**2**

votes

**0**answers

233 views

### Definite integral probably equal to zeta with known (but unusable) closed form for the indefinite integral

Related to this and
this questions.
Basically got definite integral that experimentally equals
$\zeta(s)$ both numerically and symbolically.
Closed form for the indefinite integral is known, but I ...

**12**

votes

**3**answers

1k views

### Does this infinite sum provide a new analytic continuation for $\zeta(s)$?

It is well known that the infinite sum:
$$\displaystyle \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}$$
only converges for $\Re(s)>1$.
The Dirichlet 'alternating' sum:
$$\displaystyle \zeta(s) = ...

**2**

votes

**1**answer

476 views

### Are the zeros of the sum/difference of these integrals all on the critical line?

The answers given to the question whether all zeros in the critical strip of $\zeta(s)\pm\zeta(1-s)$ lie on the critical line, suggest that this can indeed be proven, however only for those zeros ...

**1**

vote

**0**answers

169 views

### Naive conjecture about zeros and local extrema of $\Re \zeta(\sigma+i t)$ (resp. $\Im \zeta(\sigma+ it)$) for $ 0 \le \sigma \le \frac12$

Based on limited numerical evidence, I suspect this conjecture.
Conjecture: Fix $ 0 \le \sigma \le \frac12$ and let $t > 0$. Between consecutive local extrema of $\Re \zeta(\sigma+i t)$ (resp. ...

**3**

votes

**2**answers

315 views

### Consequences of a bound on possible counterexamples to Riemann hypothesis

The Riemann hypothesis has many strong consequences in number theory. The question is: would a bound on the number of zeros of Riemann zeta-function in the critical strip with real part not equal 1/2 ...

**7**

votes

**1**answer

398 views

### Is there always a zero between consecutive local extrema of $\Re \zeta(1/2+it)$ (or $\Im \zeta(1/2+i t)$

Based on limited numerical evidence, I am inclined to suspect that
there is always zero of $\Re \zeta(1/2+it)$ between consecutive local
extrema of $\Re \zeta(1/2+it)$
(and the same for $\Im ...

**0**

votes

**3**answers

254 views

### Inequality for the modulus of Riemann zeta on horizontal lines and alleged partial result of Maple

According to a conjecture p.4
$|\zeta(\frac12 -\Delta + it))| > |\zeta(\frac12 + \Delta + i t|$
for $0 < \Delta < \frac12$ and $|t| > 2 \pi +1$.
Since $\zeta(\overline{s}) = ...

**2**

votes

**1**answer

246 views

### Finite sum seemingly related to nontrivial zeta zeros

For $t \in \mathbb{R}$ define
$$ F(t) = \sum_{n=1}^{[t]} \frac{(-1)^{(n-1)}}{n^{\frac12 + it}}$$
Let $\operatorname{Arg}(t)$ be $\operatorname{atan2}(\Im t , \Re t)$ -
basically this is $\arctan$, ...

**0**

votes

**0**answers

228 views

### Differential equation for zeta on the critical line

Edit Major rewrite since Johan Andersson observed the original
question is trivial because of vanishing of coefficients.
From this question
$$ \zeta(1-x) = ...

**4**

votes

**0**answers

314 views

### $\zeta(x)$ in terms of $\zeta'(x),\zeta'(1-x),\Gamma,\psi$

By differentiating $\xi$ and solving for $\zeta(1-x)$:
$$ \zeta(1-x) = \frac{2(\zeta'(x)\Gamma(x/2)+\Gamma((1-x)/2) \zeta'(1-x)\pi^{x-1/2}) )}{\Gamma((1-x)/2) \pi^{-1/2+x}(2\log\pi ...

**2**

votes

**0**answers

199 views

### computing a certain contour integral [closed]

I want to compute an integral along a vertical line segment. The function I'm integrating involves the zeta-function, and usually the way such integrals are done treats the line segment as one side ...

**2**

votes

**0**answers

183 views

### Can $\zeta(s)$ for $\Re(s)>1$ be split into two factors that each can be analytically continued?

Assuming the RH and $s \in \mathbb{C}, \rho_n =\frac12 \pm i\gamma_n$, the following (altered) Hadamard product:
$$\displaystyle \displaystyle \prod_{n=1}^\infty \left(1- \frac{s}{\frac12+ (-1)^n i ...

**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 ...

**3**

votes

**1**answer

377 views

### How is “large” defined in an equality for the modulus of Riemann zeta?

This paper p.4 claims:
Corollary C. Assume RH. For all large $t$ we have
$$|\zeta(\frac12 +it)| \le \exp\left(\frac38 \frac{\log{t}}{\log{\log{t}}}\right) \qquad (1) $$
$t$ a Gram points often ...

**3**

votes

**2**answers

307 views

### On the critical line $ \Re \zeta'(s)/\zeta(s) =? 1/2 \log(\pi) - 1/2 \Re \psi(s/2)$ ?

For $\Re s = 1/2$ numerical evidence suggest:
$$ \Re \zeta'(s)/\zeta(s) = 1/2 \log(\pi) - 1/2 \Re \psi(s/2) \qquad (1) $$
How this was found. Consider the symmetrized zeta function
$\zeta^*(x)= ...

**41**

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 ...

**1**

vote

**0**answers

189 views

### Zeta sum $\sum_{n=2}^\infty \frac{\zeta(n)}{a^n}$

Probably this is known, but mathworld and wolfram alpha don't
recognize this potential identities.
Numerical evidence suggests:
$$ \sum_{n=2}^\infty \frac{\zeta(n)}{a^n} =? \sum_{n=1}^\infty ...

**2**

votes

**1**answer

223 views

### Sign of Mertens function

It is well-known that Mertens function $$M(N)=\sum_{i=1}^N\mu(n)$$ changes sign infinitely many times when $N\rightarrow +\infty$.
Question: Is there a proof of this statement without Riemann zeta ...

**3**

votes

**2**answers

380 views

### Bounds for $\sum_{k=1}^\infty\frac{(-1)^{k+1}x^k}{(k-1)!\zeta(2k)}$

Let $$ f(x) = \sum_{k=1}^\infty\frac{(-1)^{k+1}x^k}{(k-1)!\zeta(2k)}$$
Are there lower bounds, upper bounds or (unlikely) simpler closed form
for $f(x)$?
The bounds for Bernoulli numbers I ...

**1**

vote

**0**answers

229 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
...

**6**

votes

**1**answer

364 views

### Closed form for derivatives $\zeta^{(n)}(1/2)$

According to mathworld
41,42. "Derivatives $\zeta^{(n)}(1/2)$ can also be given in closed form"
with example for the first derivative.
What is the closed form? References?
The motivation is that ...

**3**

votes

**1**answer

375 views

### zeta(2k+1) is a rational multiple of pi^{2k} zeta'(-2 k) ?

Probably this is well know and elementary and will delete it, but couldn't find it on the web.
Got a sketch of proof and numerical evidence that
$\zeta(2k+1)$ is a rational multiple of $\pi^{2k} ...

**17**

votes

**4**answers

2k 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:
...

**3**

votes

**1**answer

374 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

**0**answers

228 views

### Alternating sums of the non-trivial zeros of $\zeta(s)$.

It is known that the infinite sum of the non-trivial zeros $\rho_n =\beta + \gamma_ni$ of $\zeta(z)$, when taken in pairs that are either conjugated or reflexive (they give the same outcome), ...

**1**

vote

**2**answers

311 views

### Does there exist a closed form for the factors of this infinite product ?

Assume $s,a \in \mathbb{C}, a \pm in \ne 0$.
The following infinite product nicely converges and can be expressed in a closed form:
$$\displaystyle \prod_{n=1}^\infty \left(1- \frac{s}{a+i n} ...

**2**

votes

**2**answers

322 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 ...

**18**

votes

**4**answers

1k views

### 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 "expectional zeros" of course first ...

**5**

votes

**2**answers

639 views

### Is there an explicit expression for the imaginary part of some non-trivial zero of zeta,

Is there an explicit expression for the imaginary part of some non-trivial zero of zeta, in terms of well-known constants, such as say $\gamma$ or $\pi$ say ?

**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) ...

**0**

votes

**1**answer

188 views

### Efficient way for computation of derivatives of $f(x) = \zeta(1-x) + 1/x $ at integer x?

[This question is copied from math.stackexchange, it didn't get answers so far]
For some exercises with (divergent) summation of the Stieltjes constants,see also MSE I'm trying a formula, which ...

**5**

votes

**1**answer

454 views

### Is there a connection between the closed forms of these two infinite products?

Take the following two infinite products that have closed forms.
Assume: $\gamma_n > 0 \in \mathbb{R};s,a,x \in \mathbb{C}; x \ne 0,a \pm ix\gamma_n \ne 0$
The first product:
$$\displaystyle ...

**3**

votes

**2**answers

581 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 ...

**15**

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 ...

**3**

votes

**0**answers

387 views

### An Expression for $\log\zeta(ns)$ derived from the Limit of the truncated Prime $\zeta$ Function

I think, here, I found
$$
P_x(s)=\sum_{p < x} \frac{1}{p^s} =\sum_{n=1}^{\infty}\frac{ \mu (n)}{n}
\sum_{z\in\{1,\rho\}}(-1)^{1-\delta_{1z}}
\left[
{\rm li}(t^{\frac zn-s})
\right]^{x}_2
\tag{7}
...

**0**

votes

**1**answer

698 views

### Can infinite polynomials be expressed as a product of its linear factors?

Background:
In the 1700s, Euler solved the Basel Problem, which was to solve $\sum_{n=1}^\infty\frac{1}{n^2}$ in closed-form. Euler showed that it was equal to $\frac{\pi^2}{6}$ by first expressing ...

**0**

votes

**0**answers

147 views

### What can be said about zeros of $\zeta(s)$ sharing the largest real part?

Specifically, if $\rho$ is such that $\zeta(\rho)=0$ and $\max_{\rho}\Re(\rho)= \Theta$, can anything interesting be said about the number/distribution of zeros on the vertical line $\sigma=\Theta$?
...

**5**

votes

**3**answers

571 views

### A closed form of infinite products of complex zeros involving $\Im(\rho_n)$. Does a proof of this closed form imply RH?

Building on this question scaling the imaginary part of $\rho$s in infinite products, I like to conjecture that:
$$\displaystyle \prod_{n=1}^\infty \left(1- \frac{s}{\mu_n} \right) \left(1- ...

**2**

votes

**1**answer

216 views

### Upper bounds for $\zeta(s)$ on the critical line

In Graham and Kolesnik's "Van der Corput's Method of Exponential Sums" they mention the results of Watt (1989) who obtained $\zeta(1/2 + it) = O(t^{89/560 + \epsilon})$.
Is anyone aware of more ...

**15**

votes

**1**answer

466 views

### 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
$$
...

**4**

votes

**4**answers

447 views

### What happens to $\zeta(s)$ when all its $\Im(\rho_n)$ are “scaled” linearly?

I found that the following infinite product with $\mu = a +n b i$ and a,b real, $s \in \mathbb{C}$:
$$\displaystyle \prod_{n=1}^\infty \left(1- \frac{s}{\mu} \right) \left(1- \frac{s}{1-\mu} ...

**1**

vote

**1**answer

306 views

### $\zeta(2k+1)$ expressed in a product of two infinite products of non-trivial zeros.

Take the Hadamard product for $\zeta(s)$:
$$\displaystyle \zeta(s) = \pi^{\frac{s}{2}} \dfrac{\prod_\rho \left(1- \frac{s}{\rho} \right) \left(1- \frac{s}{1-\rho} ...

**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 ...

**2**

votes

**0**answers

122 views

### The influence of $\chi(s)$ on complex zeros of $\frac{\zeta(\overline{s})}{\zeta(1-s)} \pm \frac{\zeta(s)}{\zeta(1-\overline{s})}$

I was exploring the formula:
$$g(s)_{\pm} := \displaystyle \frac{\zeta(\overline{s})}{\zeta(1-s)} \pm \frac{\zeta(s)}{\zeta(1-\overline{s})}$$
and found that for all $\Re(s) \ne \frac12$:
...