Questions tagged [zeta-functions]

Question 1

I would like to understand how to make sense of the following divergent series, or at least to identify the appropriate analytic continuation of its general term: \begin{align} \sum_{n=0}^{\infty}\...

Question 2

I would like to prove that $$\int_0^1 \operatorname{Li}_2 \left(\frac{1-x^2}{4}\right) \frac{2}{3+x^2} \,\mathrm dx= \frac{\pi^3 \sqrt{3}}{486}.$$ It’s known that $\Im \operatorname{Li}_3(e^{2πi/3})= \...

Question 3

I am working on a project that combines ingredients from geometric group theory and spectral graph theory, and I would appreciate references (papers, authors, or survey articles) that study anything ...

Question 4

Calculate: $${\int\limits_0^\infty{\left[{\frac{x^{s-1}}{e^x-1}-\frac{(e^x-1)^{s}}{(e^x-1)^2}}\right]dx=}{\,\,\color{red}{?}}}\tag{1}$$ For $0\lt\Re(s)\lt1$. An integral definition of $\zeta(s)$ Zeta ...

Question 5

The Airy zeta function is defined as a sum over the zeroes of the $\mathrm{Ai}$ airy function. $$\zeta _{\mathrm {Ai} }(s)=\sum _{i=1}^{\infty }{\frac {1}{|a_{i}|^{s}}}$$ Integer values for $\zeta_{\...

Question 6

Define $Z$ as a sort of zeta function defined over the powers of the non-trivial zeros $\rho$ of the Riemann zeta function (a meta-zeta function). So we have, $$Z(s) = \sum_{\rho} \frac{1}{\rho^s}$$ ...

Question 7

The Riemann zeta function has the following representations for $$\text{Re}(s) > 1$$: As a Dirichlet series: $$ \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} $$ And as an Euler product: $$ \...

Question 8

Let $$\zeta(s,x) = \sum_{n=0}^\infty (n+x)^{-s}, \qquad \Re(s)>1,\Re(x)>0$$ be the Hurwitz zeta function. As is well-known, it can be meromprhically continued to $s\in \mathbb{C}$. How to prove ...

Question 9

A week ago, a user on Quora asked this question: How do you show that $$I=\iiint_{[0,1]^{3}}\frac{\ln x\ln y\ln z\ln(1-xyz)}{(1-x)(1-y)}\mathrm{d}x\mathrm{d}y\mathrm{d}z=\frac{1}{701}\left(386\zeta(5)...

Question 10

$$\newcommand{\arccosh}{\operatorname{arccosh}} $$ I need help to proof that : $$\int^1_0 \frac{\arccos(x)}{x}\arcsin\left({\frac{\sqrt{x^2+x+1}-\sqrt{x^2-x+1}}{2}}\right)\arccosh\left({\frac{\sqrt{...

Question 11

Calculate the series $$\sum_{n,m,r>0}\frac{1}{mnr(n+m)(n+r)(m+r)}=?$$ \begin{align*} S&=\sum\limits_{n,m,r\ge1}\frac{1}{mnr}\iiint\limits_{t,u,v>0} e^{-t(n+m)}e^{-u(n+r)}e^{-v(m+r)}\,dt\,du\...

Question 12

I am trying to study the decay of a function $\Xi$ related to the Riemann's $\xi$ function. It is defined as $\Xi(t) = \xi(\frac{1}{2} + it)$, and I found that it has great relevance with the Riemann ...

Question 13

Stein's reasoning in page 170: Consider the theta function, already introduced in Chapter 4, which is defined for real $t>0$ by $$ \vartheta(t)=\sum_{n=-\infty}^{\infty} e^{-\pi n^2 t} $$ An ...

Question 14

Consider the zeta like function : $$\zeta_2(s) = \sum_{n>0} \sum_{m>0} \dfrac{1}{m n^s + n m^s}$$ It clearly converges for real $s>2$. I wonder about the analytic continuation and the poles ...

Question 15

I read somewhere that every $L$-function of an imaginary quadratic field $K$ with Grössencharacter is the $L$-function of a cusp form for a certain congruence group of $\operatorname{SL}_2(\Bbb Z)$. ...

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Questions tagged [zeta-functions]

analytic continuation of the series $\sum_{n=0}^{\infty}\frac{n^2}{\sqrt{a^2+n^2}}$

Prove that $\int_0^1\operatorname{Li}_2\left(\frac{1-x^2}{4}\right)\frac{2}{3+x^2}\,\mathrm dx= \frac{\pi^3 \sqrt{3}}{486}$

Looking for authors/papers in GGT involving piecewise isometries, free group actions, and twisted Ihara zeta functions

On $\int_0^\infty\left[\frac{x^{s-1}}{e^x-1}-\frac{(e^x-1)^s}{(e^x-1)^2}\right]dx$

Airy zeta as a polynomial

Convergence of a Zeta-Zeta function

Can a Finite Zeta Sum Be Expressed as a Finite Prime Product?

An integral of Hurwitz zeta function $\int_0^1 \zeta(\frac13,x) \zeta(\frac13,1-x)^2 dx$

Particular relation between $\pi$ and the Riemann zeta function at odd positive integers

Show that : $\int^1_0 \frac{\arccos(x)}{x} \arcsin(\cdots)\operatorname{arccosh}(\cdots) dx =\frac{\zeta^2(2)}{18}$

Calculate the series $\sum_{n,m,r>0}\frac{1}{mnr(n+m)(n+r)(m+r)}$

Asymptotic behavior of the zeta function at vertical lines in the complex plane

A problem about an interexchange of infinite sum with the integral in Stein's Complex Analysis

Analytic continuation and zeros of $\zeta_2(s) = \sum \sum \frac{1}{m n^s + n m^s}$?

Cusp forms and $L$-functions of imaginary quadradic fields

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