Questions tagged [zeta-functions]
Questions on various generalizations of the Riemann zeta function (e.g. Dedekind zeta, Hasse–Weil zeta, L-functions, multiple zeta). Consider using the tag (riemann-zeta) instead if your question is specifically about Riemann's function.
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4
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Prove that $\int_0^1\operatorname{Li}_2\left(\frac{1-x^2}{4}\right)\frac{2}{3+x^2},円\mathrm dx= \frac{\pi^3 \sqrt{3}}{486}$
I would like to prove that
$$\int_0^1 \operatorname{Li}_2 \left(\frac{1-x^2}{4}\right) \frac{2}{3+x^2} ,円\mathrm dx= \frac{\pi^3 \sqrt{3}}{486}.$$
It’s known that $\Im \operatorname{Li}_3(e^{2πi/3})= \...
1
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1
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On $\int_0^\infty\left[\frac{x^{s-1}}{e^x-1}-\frac{(e^x-1)^s}{(e^x-1)^2}\right]dx$
Calculate: $${\int\limits_0^\infty{\left[{\frac{x^{s-1}}{e^x-1}-\frac{(e^x-1)^{s}}{(e^x-1)^2}}\right]dx=}{,円,円\color{red}{?}}}\tag{1}$$
For 0ドル\lt\Re(s)\lt1$.
An integral definition of $\zeta(s)$ Zeta ...
3
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1
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Airy zeta as a polynomial
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_{\...
0
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1
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Can a Finite Zeta Sum Be Expressed as a Finite Prime Product?
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:
$$
\...
3
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2
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Particular relation between $\pi$ and the Riemann zeta function at odd positive integers
A week ago, a user on Quora asked this question:
How do you show that $$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)...
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Calculate the series $\sum_{n,m,r>0}\frac{1}{mnr(n+m)(n+r)(m+r)}$
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)},円dt,円du\...
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A problem about an interexchange of infinite sum with the integral in Stein's Complex Analysis
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 ...
0
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Cusp forms and $L$-functions of imaginary quadradic fields
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 $\operatorname{SL}_2(\Bbb Z)$. ...