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.

Riemann zeta function or, Euler–Riemann zeta function or, Zeta function in number theory are functions belonging to a class of analytic functions of a complex variable, comprising Riemann's zeta-function, its generalizations and analogues. Zeta-functions and their generalizations in the form of L-functions (cf. Dirichlet L-function) form the basis of modern analytic number theory. In addition to Riemann's zeta-function one also distinguishes the generalized zeta-function $~ζ(s,a)~$, the Dedekind zeta-function, the congruence zeta-function, etc.

Definition: The Riemann zeta function for $~s\in \mathbb{C}~$ with $~\operatorname{Re}(s)>1~$ is defined as $$\zeta(s)=\sum_{n=1}^{\infty}~\frac{1}{n^s}=1+\frac{1}{2^s}+\frac{1}{3^s}+\cdots$$ It is then defined by analytical continuation to a meromorphic function on the whole $\mathbb{C}$ by a functional equation.

Euler Product Representation: The Riemann zeta function for $~s\in \mathbb{C}~$ with $~\operatorname{Re}(s)>1~$ can be written as $$\zeta(s)=\prod_{p~\text{prime}}~(1-p^{-s})^{-1}$$

Applications: The Zeta function is an extremely important special function of mathematics and physics that arises in definite integration and is intimately related with very deep results surrounding the prime number theorem. While many of the properties of this function have been investigated, there remain important fundamental conjectures (most notably the Riemann hypothesis) that remain unproved to this day.

References:

https://en.wikipedia.org/wiki/Riemann_zeta_function

http://mathworld.wolfram.com/RiemannZetaFunction.html

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Evaluate $\int_0^1 \frac{\log \left( 1+x^{2+\sqrt{3}}\right)}{1+x}\mathrm dx$

I am trying to find a closed form for $$\int_0^1 \frac{\log \left( 1+x^{2+\sqrt{3}}\right)}{1+x}\mathrm dx = 0.094561677526995723016 \cdots$$ It seems that the answer is $$\frac{\pi^2}{12}\left( 1-\sqrt{3}\right)+\log(2) \log \left(1+\sqrt{3}…
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Gross-Zagier formulae outside of number theory

(Edit: I have asked this question on MO.) The Gross-Zagier formula and various variations of it form the starting point in most of the existing results towards the Birch and Swinnerton-Dyer conjecture. It relates the value at $1$ of the derivative…
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How to prove that $\frac{\zeta(2) }{2}+\frac{\zeta (4)}{2^3}+\frac{\zeta (6)}{2^5}+\frac{\zeta (8)}{2^7}+\cdots=1$?

How can one prove this identity? $$\frac{\zeta(2) }{2}+\frac{\zeta (4)}{2^3}+\frac{\zeta (6)}{2^5}+\frac{\zeta (8)}{2^7}+\cdots=1$$ There is a formula for $\zeta$ values at even integers, but it involves Bernoulli numbers; simply plugging it in…
E.H.E
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Evaluating the log gamma integral $\int_{0}^{z} \log \Gamma (x) \, \mathrm dx$ in terms of the Hurwitz zeta function

One way to evaluate $ \displaystyle\int_{0}^{z} \log \Gamma(x) \, \mathrm dx $ is in terms of the Barnes G-function. $$ \int_{0}^{z} \log \Gamma(x) \, \mathrm dx = \frac{z}{2} \log (2 \pi) + \frac{z(1-z)}{2} + z \log \Gamma(z) - \log…
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Does $\zeta(3)$ have a connection with $\pi$?

The problem Can be $\zeta(3)$ written as $\alpha\pi^\beta$, where ($\alpha,\beta \in \mathbb{C}$), $\beta \ne 0$ and $\alpha$ doesn't depend of $\pi$ (like $\sqrt2$, for example)? Details Several $\zeta$ values are connected with $\pi$,…
GarouDan
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Infinite Series $\sum_{n=1}^\infty\frac{H_n}{n^32^n}$

I'm trying to find a closed form for the following sum $$\sum_{n=1}^\infty\frac{H_n}{n^3\,2^n},$$ where $H_n=\displaystyle\sum_{k=1}^n\frac{1}{k}$ is a harmonic number. Could you help me with it?
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Remarkable logarithmic integral $\int_0^1 \frac{\log^2 (1-x) \log^2 x \log^3(1+x)}{x}dx$

We have the following result ($\text{Li}_{n}$ being the polylogarithm): $$\tag{*}\small{ \int_0^1 \log^2 (1-x) \log^2 x \log^3(1+x) \frac{dx}{x} = -168 \text{Li}_5(\frac{1}{2}) \zeta (3)+96 \text{Li}_4(\frac{1}{2}){}^2-\frac{19}{15} \pi ^4…
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Infinite Series $\sum_{n=1}^\infty\frac{H_n}{n^22^n}$

How can I prove that $$\sum_{n=1}^{\infty}\frac{H_n}{n^2 2^n}=\zeta(3)-\frac{1}{2}\log(2)\zeta(2).$$ Can anyone help me please?
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What is the binomial sum $\sum_{n=1}^\infty \frac{1}{n^5\,\binom {2n}n}$ in terms of zeta functions?

We have the following evaluations: $$\begin{aligned} &\sum_{n=1}^\infty \frac{1}{n\,\binom {2n}n} = \frac{\pi}{3\sqrt{3}}\\ &\sum_{n=1}^\infty \frac{1}{n^2\,\binom {2n}n} = \frac{1}{3}\,\zeta(2)\\ &\sum_{n=1}^\infty \frac{1}{n^3\,\binom {2n}n} =…
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Conjectural closed-form of $\int_0^1 \frac{\log^n (1-x) \log^{n-1} (1+x)}{1+x} dx$

Let $$I_n = \int_0^1 \frac{\log^n (1-x) \log^{n-1} (1+x)}{1+x} dx$$ In a recently published article, $I_n$ are evaluated for $n\leq 6$: $$\begin{aligned}I_1 &= \frac{\log ^2(2)}{2}-\frac{\pi ^2}{12} \\ I_2 &= 2 \zeta (3) \log (2)-\frac{\pi…
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What is a zeta function?

In my readings, I've come across a wide variety of objects called zeta functions. For example, the Ihara zeta function, Igusa local zeta function, Hasse-Weil zeta function, etc. My question is simple: What makes something a zeta function? There are…
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Factorial of infinity

So, I've read in this article that: $$\zeta'(0) = \log\sqrt\frac 1 {2\pi}$$ And that: $$e^{-\zeta'(0)} = 1\cdot2\cdot3\cdots \infty = \infty! = \sqrt{2\pi}$$ I found this result very strange and tried to simplify if but i couldn't. Can somebody help…
PPP
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A Geometric Proof of $\zeta(2)=\frac{\pi^2}6$? (and other integer inputs for the Zeta)

Is there a known geometric proof for this famous problem? $$\zeta(2)=\sum_{n=1}^\infty n^{-2}=\frac16\pi^2$$ Moreover we can consider possibilities of geometric proofs of the following identity for positive even inputs of the Zeta function: $$…
finnlim
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Motivation on how does complex analysis come to play in number theory?

I am not sure if this is a appropriate question. If it isn't, let me know and I'll delete it. $\textbf{Background}$ I am an undergraduate student and I'm very interested in number theory. I've tried to study alone some algebraic number theory and I…
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Evaluating $\prod_p(1\pm4/p^2)$ in Closed Form

Do either of the two infinite products $~\displaystyle\prod_{p~\in~\mathbb P}\bigg(1+\frac{2^2}{p^2}\bigg)~$ and $~\displaystyle\prod_{p~\in~\mathbb P}^{p~>~2}\bigg(1-\frac{2^2}{p^2}\bigg)~$ possess a closed form expression, where $\mathbb P$…
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