**1**

vote

**1**answer

231 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 + ...

**4**

votes

**1**answer

512 views

### Derivative of Riemann Zeta at nontrivial zeros

I would like to know whether the real part of the first derivative of the Zeta function at the non trivial zeros of Zeta is stricly positive and if so, is there a proof for it.
Also, are there tables ...

**5**

votes

**1**answer

469 views

### How “deep” is the unboundedness of the reciprocal of the Riemann zeta function on vertical lines in the critical strip?

I think it is probably well known that, for every $1/2<\sigma\leq 1$, the function $1/\zeta(\sigma+it)$ is unbounded.
Yet, I cannot decide how deep this is. I imagine it could be proved using a ...

**2**

votes

**1**answer

283 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 ...

**16**

votes

**4**answers

990 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 ${\rm Re}(s) = 1$...

**2**

votes

**1**answer

484 views

### On the convergence of Dirichlet series over the Mobius Mu function

It is known that if $\sum_{k=1}^{\infty} \frac{\mu(k)}{k^s} = \frac{1}{\zeta(s)}$ for $\Re(s) > 1/2$ then RH holds. My question is:
Under RH why is it not $\sum_{k=1}^{\infty} \frac{\mu(k)}{...

**6**

votes

**0**answers

305 views

### implication of divergence of $1/\zeta(s) $ at 1/2

$1/\zeta(s)=\sum_{n>0}\frac{\mu(n)}{n^s}$ where $\mu$ is the Moebius function. This series is known to converge for $s\ge 1$ and diverge for $s\le 1/2$.
Its convergence is unknown if $1/2< s<...

**5**

votes

**2**answers

637 views

### On extended Riemann Hypothesis and coefficients of Selberg Class L-functions

There is the conjecture that Selberg Class L-functions satisfy RH.
So that an L-function needs to have its coefficient multiplicatives (plus other conditions: functional equation,...) in order to ...

**0**

votes

**2**answers

309 views

### What is known about the set $S$ of couples of rationals $(q,q')$ such that $\zeta(q+iq')$ is rational?

The question is the title. For example, if we could show that $S$ is finite, then this would entail that every large enough integer $n$ is such that $\zeta(2n+1)$ is irrational and that, under RH, ...

**4**

votes

**1**answer

224 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 ...

**6**

votes

**1**answer

191 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

250 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

255 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 {s-1-2\,n}{{n}^{s}}...

**15**

votes

**2**answers

2k 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)...

**3**

votes

**1**answer

344 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

266 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 ...

**15**

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

501 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

183 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

374 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

448 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 \zeta(1/...

**0**

votes

**3**answers

276 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}) = \overline{\...

**2**

votes

**1**answer

283 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 $\arctan$, ...

**6**

votes

**0**answers

356 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 -\psi((1-x)/2)-\...

**2**

votes

**0**answers

219 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 ...

**5**

votes

**1**answer

340 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

234 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

388 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

361 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)= \pi^{...

**49**

votes

**5**answers

5k 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

287 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 \frac{1}...

**2**

votes

**1**answer

264 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 ...

**5**

votes

**2**answers

523 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 ...

**6**

votes

**0**answers

498 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/...

**7**

votes

**1**answer

429 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

483 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} \...

**18**

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

481 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

272 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

363 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} \right)...

**2**

votes

**2**answers

376 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 ...

**21**

votes

**4**answers

2k 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 "exceptional zeros" of course first ...

**6**

votes

**2**answers

701 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 ?

**32**

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) zeta-...

**0**

votes

**1**answer

205 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

503 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 H_{...

**3**

votes

**2**answers

652 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 ...

**6**

votes

**0**answers

511 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}
$...

**2**

votes

**1**answer

1k 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 $\...