# Tagged Questions

**5**

votes

**1**answer

163 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 ...

**2**

votes

**1**answer

102 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

665 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 ...

**1**

vote

**0**answers

161 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}} + ...

**0**

votes

**0**answers

102 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

71 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 ...

**7**

votes

**1**answer

528 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.

**8**

votes

**1**answer

177 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 ...

**23**

votes

**1**answer

854 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

363 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

71 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

833 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

424 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 ...

**1**

vote

**0**answers

275 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}}} + ...

**4**

votes

**1**answer

409 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

351 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

**0**answers

156 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 ...

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votes

**4**answers

895 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) = ...

**0**

votes

**0**answers

232 views

### New formula for polylogarithm and the fastest converging series for $\zeta(3)$?

Recently I found a formula for the infinite hyperbolic sine and cosine series also described in a .pdf here.
I write them as polylogarithms, as they are already extended by the process of analytic ...

**2**

votes

**1**answer

259 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

266 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

566 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

303 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, ...

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votes

**0**answers

80 views

### Closed forms for paired factors of a 4-factor infinite product. Is their shape fixed?

As a follow up on this question, I have now composed the following model of closed forms for infinite products of pairs of factors. At the heart is a 4-factor infinite product shown in red at the ...

**4**

votes

**1**answer

205 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 ...

**1**

vote

**0**answers

192 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

199 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 ...

**3**

votes

**1**answer

310 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

230 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

465 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

163 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

311 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 ...

**6**

votes

**1**answer

391 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

248 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

239 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

213 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

308 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

198 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

175 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

195 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

371 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

300 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)= ...

**39**

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

177 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

218 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

374 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

**1**answer

355 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

359 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

1k 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:
...