**5**

votes

**3**answers

343 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

66 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 ...

**23**

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

132 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 ...

**0**

votes

**1**answer

415 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 ...

**26**

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 ...

**5**

votes

**0**answers

203 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

**2**answers

144 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 $\Xi(z)$ ...

**3**

votes

**1**answer

366 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

234 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

196 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

51 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

178 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

104 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

711 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

163 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}} + ...

**7**

votes

**1**answer

534 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

110 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

226 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 ...

**2**

votes

**1**answer

284 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

187 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

85 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

874 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

154 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

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

837 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

433 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

185 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

319 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

300 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

278 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 ...

**40**

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

276 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

**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, ...

**5**

votes

**2**answers

625 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 ?

**1**

vote

**0**answers

147 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 ...

**4**

votes

**1**answer

414 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

360 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 ...

**14**

votes

**4**answers

899 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) = ...

**9**

votes

**2**answers

627 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 ...

**2**

votes

**0**answers

159 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 ...

**6**

votes

**2**answers

1k views

### Riemann zeta function at positive integers and an Appell sequence of polynomials related to fractional calculus

I was exploring some raising and lowering operators related to an infinitesimal generator for fractional integro-derivatives and found an Appell sequence of polynomials, i.e., an infinite sequence of ...

**9**

votes

**1**answer

1k views

### The Riemann's Zeta Function represented as a continued fraction and a question of convergence.

The Riemann's zeta function can be expressed as a continued fraction as follows
\begin{align*}
\zeta(z)=\newcommand{\bigk}{\mathop{\Huge\vcenter{\hbox{K}}}}\left(1-\bigk_{k=1}^{\infty ...

**0**

votes

**0**answers

238 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 ...

**1**

vote

**0**answers

213 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
...

**2**

votes

**1**answer

262 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

269 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< ...