**0**

votes

**1**answer

180 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.
...

**23**

votes

**1**answer

978 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 ...

**1**

vote

**1**answer

659 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

73 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

871 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

473 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

194 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

**0**answers

364 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

320 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 ...

**1**

vote

**0**answers

295 views

### Do these infinite series expressing $\zeta(s)$ only (partially) converge at $\Re(s)=\frac12$?

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

**1**

vote

**0**answers

171 views

### Some identities with the Riemann-Hurwitz zeta function

The only definition that I have ever seen of this Riemann-Hurtwitz zeta-function is this,
For $0 < a \leq 1$ we have the identity
$$ \zeta(z, a) = \frac{2 \Gamma(1 - z)}{(2 \pi)^{1-z}} \left[\sin ...

**1**

vote

**2**answers

194 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

437 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

388 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 ...

**3**

votes

**1**answer

252 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 ...

**14**

votes

**4**answers

934 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) = ...

**2**

votes

**1**answer

320 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} ...

**6**

votes

**0**answers

288 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< ...

**5**

votes

**2**answers

594 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 ...

**1**

vote

**2**answers

304 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

210 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

172 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

212 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

211 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

911 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

321 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

244 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 ...

**14**

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

478 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

174 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

327 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

411 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

259 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

254 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

253 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) = ...

**6**

votes

**0**answers

334 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

203 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

194 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

205 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

381 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

315 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)= ...

**43**

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

215 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

232 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

390 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

260 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

379 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

400 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

390 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 ...