**2**

votes

**2**answers

223 views

### Trivial zeroes of the Riemann Zeta function are simple

The trivial zeroes of the Riemann Zeta function are located on $-2\mathbb N^*$ and they are simple. It is not difficult to see that, but the proof I have in mind is using the fact that ...

**-4**

votes

**0**answers

72 views

### Non-standard numbers and exponential form of Zeta function

Basic idea
For a long time I was looking for a numerical system that would allow to compare infinite sets. In contrast to Cantor's approach that empathizes the possibility of on-to-one correspondence ...

**-2**

votes

**0**answers

63 views

### Are there papers or books that explain why Bernhard Riemann believed that his hypothesis is true?

This question accross this question from SE such that no answer ,I think every
mathematician should know Wy B.Riemann believed that his hypothesis is true .I'm affraid
to ask this question in M.O ...

**-4**

votes

**1**answer

175 views

### Calculating Riemann Non-Trivial Zeros [closed]

I recently started studying about the Riemann Hypothesis. I just want to understand if there is any condition that the imaginary part i.e. t in (0.5 + it) should always be greater than 1? For Example, ...

**2**

votes

**2**answers

220 views

### Approximations to the Mertens function

The Mertens function $M(x)$ is the summatory Möbius function i.e.
$$M(x) = \sum_{k=1}^{x} \mu (k)$$
The conjecture that $M(x) = \mathcal{O}\left(x^{\frac{1}{2} + \epsilon}\right)$ was shown to be ...

**10**

votes

**0**answers

427 views

### One-to-one correspondance between zeta zeros and the prime powers? [closed]

This question is highly speculative, but I would really appreciate some insight into the problem. Previously asked on MSE without answer here.
I have noticed an interesting property related to the ...

**3**

votes

**1**answer

281 views

### Is this differential equation for zeta on the critical line? One can compute it from its derivative and simpler functions

Looks like on the critical line one can compute
$\zeta(1/2+it)$ from $\zeta^{'}(1/2+it)$ and simpler functions.
Let
$$
\begin{aligned}
f(t)= & 2\, \left( {\frac { \left( \left| \zeta^{'} ...

**4**

votes

**0**answers

143 views

### Are there infinitely many zeros of $\chi(s)+ \dfrac{2^{s}- 2^{2s-1}}{2^{s}-1}$ on the critical line?

Take $\chi(s)= 2^s\,\pi^{s-1}\,\sin\left(\frac{\pi\,s}{2}\right)\,\Gamma(1-s)$, so that $\zeta(s)=\chi(s)\,\zeta(1-s)$.
The zeros of $\chi(s)=-1$ and the non-trivial zeros $\rho$ of $\zeta(s)$, seem ...

**3**

votes

**0**answers

99 views

### Do the roots of this equation involving two Euler products all reside on the critical line?

This question loosely builds on the second part of this one.
Take the Riemann $\xi$-function: $\xi(s) =\frac12 s\,(s-1) \,\pi^{-\frac{s}{2}}\, \Gamma\left(\frac{s}{2}\right)\, \zeta(s)$. Numerical ...

**0**

votes

**0**answers

54 views

### Function series involving a suite of imprimitive Dirichlet characters and a zero of $L(\chi,s)$

I have difficulties to understand the behavior of following suite of functions near zero :
$$F_P(x)= \sum\limits_{n=P}^{\infty} \chi_P(n) f(nx)$$
With $f(x) = x^{-s_0} e^{-x}$ where $s_0$ is a non ...

**2**

votes

**0**answers

84 views

### Are all complex zeros of $\Psi^{(s)}(1) \pm \Psi^{(1-s)}(1)$ on the critical line?

veThe balanced polygamma function $\Psi^{(s)}(x)$ for $x=1$ can be expressed as:
$$\Psi^{(s)}(1)=\dfrac{\big(\Psi(-s)+\gamma\big)\,\zeta(s+1)+\zeta'(s+1)}{\Gamma(-s)}$$
Note $\Psi(s)$ is the digamma ...

**5**

votes

**1**answer

235 views

### Are all complex zeros of $\dfrac{\zeta'}{\zeta}(s) \pm \dfrac{\zeta'}{\zeta}(1-s)$ on the critical line $\Re(s)=\frac12$?

Numerical evidence suggests that all complex zeros (real ones exist as well) of:
$$\frac{\zeta'}{\zeta}(s) \pm \frac{\zeta'}{\zeta}(1-s)$$
reside on the critical line with $\Re(s)=\frac12$.
I made ...

**1**

vote

**1**answer

156 views

### Counting prime powers $p^{\frac{k}{t}} \left( t \in \mathbb{R}{+}, k \in \mathbb{N} \right)$ by changing $\rho$'s in $\psi(x)$?

With $\rho=\beta+ \gamma \,i$ being a non-trivial zero of $\zeta(s)$, the logarithmic prime counting function is:
$$\psi(x) = x - \log(2\pi) - \frac12 \log\left(1- \frac{1}{x^2}\right) - ...

**2**

votes

**0**answers

56 views

### how understand periodicity in a combination of power, gamma and zeta functions?

Riemann's functional equation may be written:
$$
\frac{\zeta(s)}{\zeta(1-s)} = 2^s \pi^{s-1} \sin(\frac{\pi s}2) \Gamma(1-s) \tag{1}
$$
and so by symmetry:
$$
\frac{\zeta(1-s)}{\zeta(s)} = 2^{1-s} ...

**8**

votes

**1**answer

376 views

### Is this theorem on $L$-functions known?

Notations For $f$ a meromorphic function on a domain $\Omega\subseteq \textbf{C}$, we shall say for convenience that $f$ is represented by an Ordinary Dirichlet Series (ODS) if $f$ can be written ...

**4**

votes

**1**answer

224 views

### The horizontal distribution of zeros of $\zeta^\prime(s)$

I have a question about a detail in the proof of Proposition 1.6 in "The horizontal distribution of zeros of $\zeta^\prime(s)$", K. Soundararajan, Duke J. Math. vol. 91 1998.
Throughout I will ...

**6**

votes

**0**answers

127 views

### How to assess the influence of a specific term in this telescoping series for $\zeta(s)$?

I like to expand on this (unanswered) MSE question.
Take the following, nicely symmetrical, telescoping series for $\zeta(s)$:
$$\displaystyle \zeta(s) = \frac{1}{2\,(s-1)} \left(1+\sum ...

**6**

votes

**2**answers

301 views

### Asymptotic expansion of $\zeta(s \mid a,b)= \sum_{n=1}^{\infty} \frac{1}{(n+a)^{s}(n+b)}$

I'm interested in an asymptotic expansion of the following Riemann zeta-type function
$$
\begin{align}
\displaystyle \zeta(s \mid a,b) := \sum_{n=1}^{\infty} \frac{1}{(n+a)^{s}(n+b)},
\quad \Re a ...

**4**

votes

**1**answer

304 views

### Square-free grows as $6n/\pi^2$: $k$-th free?

The asymptotic number of
square-free numbers
$\le n$ is $Q(n) = 6n/\pi^2 + O(\sqrt{n})$.
Because
$\zeta(2)=\pi^2/6$,
$Q(n) \approx n/\zeta(2)$.
OEIS A004709
says that cube-free numbers have ...

**2**

votes

**0**answers

133 views

### Are all zeros of $\xi(a\,s) \pm \xi\left(a\,(1-s)\right)$ on the critical line for $\forall a \in \mathbb{R}/0$?

This question expands on this one and seems to have a stronger result.
Take the Riemann $\xi$-function $\xi(s) =\frac12 s\,(s-1) \,\pi^{-\frac{s}{2}}\, \Gamma\left(\frac{s}{2}\right)\, \zeta(s)$. We ...

**4**

votes

**1**answer

580 views

### How good can we approximate with algebraic curve the egg shaped vanishing of $\Re \zeta(s)$ near the origin?

Related to this question
where degree $2$ algebraic curve is good approximation to vanishing of
the real part of expression involving zeta.
Near the origin, $\Re \zeta(s)$ vanishes in egg shaped ...

**6**

votes

**0**answers

116 views

### How comes vanishing of the real part of function involving zeta is very well approximated by algebraic curve?

In this question
Agno asked about the zeros of $\zeta\left(\frac{s}{a}\right) \pm \zeta\left(\frac{1-s}{a}\right)$.
I fixed $a=2$ and the minus sign and defined:
$$
f(s)=\Re \left( ...

**0**

votes

**0**answers

186 views

### Does the Euler product converge at $s=1$ for the Dirichlet $L$ function?

For the Riemann Zeta function, the Euler product converges on $\{Re(s)=1\}$ except at $s=1$.The zeta series diverges everywhere on $\{Re(s)=1\}$. But the $L$ series converges on $\{Re(s)>0\}$. What ...

**2**

votes

**0**answers

58 views

### Are the complex zeros of $\zeta\left(\frac{s}{a}\right) \pm \zeta\left(\frac{1-s}{a}\right)$ all on the critical line for $a \lt 0, a \ge 1$?

With $s \in \mathbb{C}, a \in \mathbb{R}$,
numerical evidence strongly suggests that the complex zeros in the critical strip of:
$$\zeta\left(\frac{s}{a}\right) \pm ...

**1**

vote

**0**answers

221 views

### On a property of Riemann Zeta function zeros

Lets consider the function : $$F(x) = \sum_{n=1} (xn)^{-s_0} e^{-nx} $$
with $s_0$ a zero of the Riemann Zeta function in the critical strip.
This sum is well defined for $x \in \mathbb{R}^{+*}$. It ...

**9**

votes

**0**answers

274 views

### Do all complex zeros in the strip of $\frac{\zeta(s)}{\Gamma(s)} - \frac{\Gamma(1-s)}{\zeta(1-s)}$ lie on the critical line?

Numerical evidence suggests that the complex zeros of:
$$f(s):=\frac{\zeta(s)}{\Gamma(s)} - \frac{\Gamma(1-s)}{\zeta(1-s)}$$
all reside on the line $\Re(s)=\frac12$, except for a finite few outside ...

**4**

votes

**0**answers

219 views

### A relation between the Gamma function and the Mobius function?

It is well known how altering the integral for the Gamma function:
$$\displaystyle \Gamma(s) = \int_0^\infty t^{s-1} e^{-t}\,dt$$
through substituting $t=nx$,
$$\displaystyle \Gamma(s)\frac{1}{n^s} ...

**5**

votes

**0**answers

121 views

### Are these identities Newton series?

Newton series is the following expansion of a function:
$$f(x)=\sum_{k=0}^\infty \binom{x}k \Delta^k [f]\left (0\right)=\sum_{n=0}^{\infty} {x\choose n} \sum_{k=0}^n{n\choose k}(-1)^{k-n}f(k)$$
Now ...

**4**

votes

**0**answers

206 views

### Telescoping series for $\zeta(s)$, question about the basic ideas and a specific series

There are many known telescoping series that enable analytic continuation of $\sum _n \frac {1}{n^{s}}$ into a variety of domains, however they seem to all be derived from two basic ideas:
1) The ...

**0**

votes

**0**answers

82 views

### Arithmetic functions associated with Hurwitz Zeta function raised to arbitrary complex powers, $\zeta(s,q)^z$ for $q \in \mathbb{N}$?

If $\zeta(s)$ is the Riemann Zeta function, then $\zeta(n)^z$, with $z \in \mathbb{C}$, $\Re(s)>1$, can be represented as
$$\zeta(s)^z=\sum_{n=1}^\infty \frac{d_z(n)}{n^{-s}}$$
where $d_z(n)$ ...

**5**

votes

**0**answers

713 views

### Zeta function double product

Is it possible to write the following double product in terms of the zeta function?
\begin{align}
&\prod_{i=1}^{\infty}\prod_{j=1}^{\infty} \frac{1}{1-(p_i\ p_j)^{-s}}
\end{align}
Extending the ...

**12**

votes

**4**answers

670 views

### Zeros of the derivative of Riemann's $\xi$-function

The Riemann xi function $\xi(s)$ is defined as
$$
\xi(s)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s).
$$
It is an entire function whose zeros are precisely those of $\zeta(s)$.
Since $\xi$ is real ...

**1**

vote

**1**answer

225 views

### Residues and values of Riemann Zeta function at some points

I need the following computational results for proving something.
Let $1/2 + i\gamma_0$, be the first nontrivial zero of Riemann zeta function, $\zeta(s)$,
i.e. $\gamma_0\sim 14.134...$.
1) what is ...

**32**

votes

**1**answer

2k views

### $\zeta(0)$ and the cotangent function

In preparing some practice problems for my complex analysis students, I stumbled across the following. It is not hard to show, using Liouville's theorem, that
$$\pi\cot(\pi ...

**1**

vote

**0**answers

133 views

### Simultaneous vanishing of convolutions of Mertens function with itself

By Landau's theorem on Dirichlet series, we know that all the step functions ($k\geq 1$)
$$M_k(x)=\frac{1}{2\pi i}\int^{2+i\infty}_{2-i\infty}\frac{x^sds}{\zeta^k(s)s}=\sum_{n\leq ...

**7**

votes

**2**answers

398 views

### Visibility interpretation of Riemann zeta zeros on the critical line?

This is a long shot, but ...
The fraction of $\mathbb{Z}^2$ lattice points
visible from the origin
$1/\zeta(2)=6/\pi^2 \approx 61$%.
The fraction of $\mathbb{Z}^3$ lattice points visible
from the ...

**11**

votes

**0**answers

1k views

### The zeta function and classical mechanics

In this paper, Guilherme França and André LeClair show that $$\gamma_{y}\sim 2 \pi \left(y-11/8\right)/W\left((y-11/8)e^{-1}\right)$$ where $W$ is the Lambert W function, and $\gamma_{y}$ is the ...

**10**

votes

**3**answers

835 views

### Why do $12$ and $120$ occur very often in the denominators of $\zeta(-n)$ for odd $n$?

$\zeta(-n) = - \dfrac{B_{n+1}}{n+1}$
$\zeta(-2n) = 0$
$\zeta(-1) = - \dfrac{1}{12}$
$\zeta(-3) = \dfrac{1}{120}$
$\zeta(-5) = - \dfrac{1}{252}$
$\zeta(-7) = \dfrac{1}{240}$
$\zeta(-9) = - ...

**3**

votes

**1**answer

578 views

### what would be the consequences on the distribution of primes of $\Lambda=\infty$?

It is widely believed that the quantity $\Lambda:=\lim\sup\dfrac{t_{n+1}-t_{n}}{2\pi/\log t_{n}}$, where $t_{n}$ is the imaginary part of the $n$-th non-trivial zero on the critical line of the ...

**1**

vote

**0**answers

92 views

### Conjectured alternate form for vanishing of $\Re\zeta(1/2+it)$ except at zeros

Heavily based on Agno's question.
Define:
$$ \chi(s)=\pi^{-(\frac{s}{2})} \Gamma(\frac{s}{2}) $$
Agno conjectured: only for $\sigma=\frac12$, $\Re(\chi(s)) = \Re(\zeta(s)) =0$ is always true, ...

**1**

vote

**0**answers

136 views

### Except for a finite few outside the strip, do all complex zeros of $\zeta(a+s)\pm \zeta(a+1-s)$ reside on the critical line for all $a\lt 0$?

Assume $a \in \mathbb{R}$ and $s \in \mathbb{C}$.
Numerical evidence suggests that all complex zeros, except for a finite few outside the strip, of:
$$\zeta(a+s)\pm \zeta(a+1-s)$$
lie on the line ...

**7**

votes

**0**answers

212 views

### Cesaro summation of a particular Dirichlet series associated with $\zeta(s)$

If you've investigated the error in Perron's formula in general, you've probably noticed that Cesaro summation $$\lim_{x\rightarrow\infty}\sum_{n\leq x} \left(1-\frac{\log n}{\log ...

**0**

votes

**0**answers

60 views

### $\eta(s)$ expressed as an 'alternating' sum of Hurwitz Zetas. Why does it only work for sums with an even number of terms?

It is known that:
$$\zeta(s)= a^{-s}\,\sum_{k=1}^{a} \zeta_H\left(s,\frac{k}{a}\right)$$
is valid for all $a \in \mathbb{N}$ and all $s \in \mathbb{C}\,/1$, with $\zeta_H$ being the Hurwitz zeta ...

**3**

votes

**2**answers

367 views

### Conjectured relation between alternating Prime zeta series and Riemann zeta

Let $P(s)$ be the Prime zeta function.
Numerical evidence suggests these identities:
$$ \sum_{k=1}^\infty \frac{(-1)^{k}P(3k)}{k}=\log{\bigg(\frac{1}{945}\frac{\pi^6}{\zeta(3)}\bigg)}\qquad\quad ...

**12**

votes

**2**answers

609 views

### Special values of $\zeta$ outside the real line and the critical strip

The values of Riemann's function at the integers have been extensively studied. I was wondering, is there anything interesting known (or conjectured) to happen arithmetically outside the real line ...

**0**

votes

**1**answer

127 views

### Estimating the height required to find a given small value of $|\zeta(s)|$ near the line $\sigma=1$

There are some qualitative theorems of Bohr, Jessen and Titchmarsh (e.g. The Theory of the Riemann zeta function, E.C. Titchmarsh, pages 306-308) proving that there is a $K=K(a,\alpha,\beta)$ such ...

**3**

votes

**1**answer

162 views

### On the domain of convergence of usual definition of Riemann zeta function [closed]

If we define Riemann zeta function as it is usually defined so that $$\zeta (s)=\sum_{n=1}^{\infty}n^{-s}=\sum_{n=1}^{\infty}e^{-s\ln n}$$ then we can rewrite it as:
$$\zeta ...

**1**

vote

**0**answers

87 views

### Are all complex zeros of $Li_s(i)\, + \, Li_{1-\overline{s}}\,(-i)$ equal to the $\rho$'s?

Take the well known square relationship for polylogarithms:
$$Li_s(z)\, + \, Li_{s}(-z)=2^{1-s}Li_s(z^2)$$
Assume $z=i$:
$$Li_s(i)\, + \, Li_{s}(-i)=2^{1-s}Li_s(-1)=-2^{1-s}\,\eta(s)$$
with ...

**1**

vote

**0**answers

85 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 ...

**1**

vote

**0**answers

163 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 ...