# Questions tagged [zeta-functions]

Questions on the various generalizations of the zeta function of Riemann. Consider using the tag (riemann-zeta) instead if your question is specifically about Riemann's function.

465 questions
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### Prove $\sum_{n=1}^\infty \frac{\zeta(2n+1) - 1}{(2n+1) 2^{2n}} = 1 + \ln(2) - \ln(3) - \gamma$

I've found the following series on the Wikipediapage of the Euler-Mascheroni constant and I want to prove it. $$\sum_{n=1}^\infty \frac{\zeta(2n+1) - 1}{(2n+1) 2^{2n}} = 1 + \ln(2) - \ln(3) - \gamma$$...
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### On the functions $\mathrm{Gi}_{s}^{p,q}(x)=\sum\limits_{n\geq0}\frac{x^{pn+q}}{(pn+q)^s}$

I have stumbled across the functions $$\mathrm{Gi}_s^{p,q}(x)=\sum_{n\geq0}\frac{x^{pn+q}}{(pn+q)^s}$$ And I would like to know where I can learn more about them. These functions are interesting ...
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### A taylor series for $\lim \limits_{h \to 0} \frac{\zeta'(-2n-h)}{\zeta(-2n-h)}$ [on hold]

Please help and use Big O Notation, thanks. I tried so-far making a Taylor Series for this limit but my solution is not what I seek for.
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### Why is $res(\Gamma(s)x^{-s}\zeta(2s)|s=\frac{1}{2}) = \frac{\Gamma(\frac{1}{2})}{2x^{\frac{1}{2}}}$?

Why is $res(\Gamma(s)x^{-s}\zeta(2s)|s=\frac{1}{2}) = \frac{\Gamma(\frac{1}{2})}{2x^{\frac{1}{2}}}$? where res means the residue of the function? I know $\zeta(s)$ has a pole at $s=1$ but i can't ...
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### Prove the series $\sum n^{-1-it}$ is diverge for all real $t$.

Prove that the series $\sum_{n=1}^\infty n^{-1-it}$ diverges for all real $t$. I have shown in the previous exercise that this series is bounded for nonzero $t$, and when $t=0$, it is famous that the ...
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### Alternating Bertrand series

It is known that $$\frac{\partial^n\zeta(s)}{\partial s^n}=(-1)^n\sum_{k=1}^\infty{\frac{\log^nk}{k^s}}$$ Can the following alternating version of the sum be expressed in terms of well-known ...
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### zeta distribution when s->1

Zeta distribution is just a set's density when s->1. I found this on the Wikipedia page about zeta distribution(https://en.wikipedia.org/wiki/Zeta_distribution). but I can't find a proof for it, can ...
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### Expected value of zeta distribution [closed]

For what values of s, expected value of zeta distribution is finite? I know that the answer is s>2, but I'm not sure how to get there.
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### Proof of Symmetry of zeros in critiical strip

How can we prove that zeros in critical strip are symmetric respect line $\sigma=\frac{1}{2}$ and respect real axis?
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### Calculating Value of $\pi$ using Zeta Function on C++ spits error

I'm not sure if this is the correct place to ask this, but surely it's not a programming question. $$\zeta (s) = \prod \limits_{p \space prime} \left ( 1 - \frac{1}{p^s} \right)^{-1}$$ So, as we ...
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### Is this zeta-type function meromorphic?

In An older question I asked : ( See A Thue-Morse Zeta function (Generalized Riemann Zeta function and new GRH) ) —— Consider $t_n$ as the Thue-Morse sequence. Let $m$ be a positive integer and $s$ ...
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### Calculating $S=\sum\limits_{n=1}^\infty\left(\frac{1}{\Gamma^2(n+1)}\right)^{{1}/{n}}$

I tried to find the answer for the question: Numerical evaluation of $\sum_{N=1}^\infty\left(\frac{1}{\Gamma(N+1)^2}\right)^{\frac{1}{N}}$. I think my result is $4$ times than the expected value. Is ...
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### A computation in Conrey's paper on Riemann zeta function

I am reading the Conrey's paper "More than two fifths of the zeros of the Riemann zeta function are on the critical line" (see here). I have question/doubt in a particular step: In P.10, it claimed ...
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### Error in expression of incomplete zeta function?

Background & Question I realised I could do a different manipulation from the one I did over here Strange method to obtain strange number theoretic identities? However after making some ...
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### Limit of local minimum of function involving zeta function ($e\log x$)

I came across the following interesting limit $$\lim\limits_{n\to\infty}\left(\min(x^{n+1}\zeta(x))-\min(x^{n}\zeta(x))\right)$$ for $x\in\mathbb{R}$ and $x\gt1.$ Calculation highly suggests the ...
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### Order of $\Gamma(n/2)$

I want to estimate the order of $\Gamma(n/2)$. We have from Stirling interpolation , for sufficiently large value of $n$, \begin{align} \Gamma(n/2)&\approx \sqrt{\frac{4\pi}{n}}\left(\frac{n}{2e}...
Prove that $\displaystyle \zeta(s)=\frac{1}{s-1}+\gamma+O(s-1)$,near $s=1$, where $\gamma$ is Euler's Constant. I've proved $\displaystyle \zeta(s)=s\int_1^{\infty}\frac{[x]-x+1/2}{x^{s+1}}\,dx+\frac{... 0answers 27 views ### Superscript numbers on a Hurwitz Zeta function (Wolfram Alpha) http://www.wolframalpha.com/input/?i=x(n%2B1)%3Dsqrt(n%2F(n%2B2))*x(n) See the x(n) solution given - there is a Zeta function with a (0,1) superscript, which I'm unfamiliar with. Additionally, the ... 1answer 48 views ### How to prove that$3^{x^2+x} (x+1)^{-x} \Gamma (x+1)\ge 1$for$x>0$? Let $$f(x)=3^{x^2+x} (x+1)^{-x} \Gamma (x+1).$$ Drawing a picture with any computer algebra system, it is obviously that$f(x) \ge 1$on$[0,\infty)$. But How can we prove this? If we take ... 1answer 115 views ### How to show that$\int_0^1 \sin \pi t ~ \left( \zeta (\frac12, \frac{t}{2})-\zeta (\frac12, \frac{t+1}{2}) \right) dt=1$? I've been trying to prove Fresnel integrals by real methods and encountered an interesting problem. Let's start with the known result: $$\int_0^\infty \sin y^2 dy = \sqrt{\frac{\pi}{8}}$$ Can we ... 1answer 40 views ### Alternating series of 'zeta' functions for real integer parameters equals 1/2 I would like to know how to show that: $$\sum_{s=2}^{\infty}(-1)^s\zeta(s)=\dfrac{1}{2}$$ where $$\zeta(s)=\sum_{n=1}^{\infty}\dfrac{1}{n^s} \\s,n \in N$$ Kind regards 1answer 42 views ### How Riemann made Z(s) to converge for s>0? I am going through the procedure that Riemann took to expand the Euler's Z-funtion. But I can not understand first part of it. We know that Z(s) just converge for s>1. How it is possible to make ... 0answers 42 views ### A question about the rationality of Hurwitz Zeta functions. I'm checking over some of my work from studying prime numbers, and I found where I used the Hurwitz Zeta function$$\zeta(s,\alpha)_{\alpha \in [0,1)} = \sum_{n\in \mathbb{N}} \frac{1}{(n+\alpha)^s}, ... 0answers 91 views ### Deriving the closed form of Gamma function using Euler-Chi function Background #1 Here is a part of an answer of @Sankyu Kim in MathOverflow. Consequently, we get the Euler-chi function$\chi(z):=\frac{\zeta(1-z)}{\zeta(z)}$. And I want to know if Sankyu Kim's ... 0answers 29 views ### Epsteins zeta function The Epstein's zeta function is:$Z(Y,s)=\underset{0\neq a\in\mathbb{Z}^{n}}{\sum}(a^{t}Ya)^{-s}$, where Y is a positive symmetric difinite$n \times n$matrix. Why does it converges when$\sigma > ...
This is Exercise 23(b) of Chapter V (Algebraic Extensions) from Lang's Algebra. Let $k$ be finite field with $q$ elements, and let $\pi_q(n)$ be the number of monic irreducible polynomials \$p \in k[...