# 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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### "Are there any simple groups that appear as zeros of the zeta function?" by Peter Freyd; why is this consternating to mathematicians?

I would like to understand the "upsetting"-to-mathematicians nature of this question Freyd poses to demonstrate that "any language sufficiently rich that to be defined necessarily ...
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### Estimates of the derivatives of $\Xi(s)$

The $\Xi$ Function is defined by $\Xi(s)=\xi(\frac{1}{2}+is)$, where $\xi(s)=\frac{1}{2}s(s-1)\pi^{-\frac{s}{2}}\Gamma(\frac{s}{2})\zeta(s)$. This is a problem from my homework: since we can write it ...
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### Relationship between $\zeta(3)$ and ordinary logarithm function $\text{log}(\text{x})$

While working on another problem, I came up with the following expression. This involved many manual definite integral evaluations and not at all elegant. So I am not going into the details of the ...
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### Evaluating the sum $\sum_{n=1}^\infty \frac{1}{n(n+a)^b}$ [duplicate]

I am looking for ways to simplify the sum $$\sum_{n=1}^\infty \frac{1}{n(n+a)^b}, \quad a\in\mathbb{R}^+, b\in\mathbb{N}.$$ The first thought I had approaching this was to use Hurwitz and/or Zeta ...
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I've been playing around with some integrals, and I started looking at Apery's constant. There are some integral representation I've found online, such as: $$\zeta(3)=\frac{16}{3}\int_{0}^{1}\frac{x\... • 71 4 votes 3 answers 136 views ### I need help evaluating the integral \int_{-\infty}^{\infty} \frac{\log(1+e^{-z})}{1+e^{-z}}dz I was playing around with the integral:$$\int_{-\infty}^{\infty} \frac{\log(1+e^{-z})}{1+e^{-z}}dz$$I couldn't find a way of solving it, but I used WolframAlpha to find that the integral evaluated ... • 71 0 votes 0 answers 59 views ### What are these points of lower magnitude in the Riemann Zeta? I recently visualized the Riemann Zeta function and noticed an interesting pattern: there are points of slightly lower magnitude along a very slightly curved vertical line, extending furthest out to ... 1 vote 0 answers 50 views ### A problem with Mertens Theorem According to Mertens Theorem$$ \lim_{n \to \infty } (\frac{1}{ln(p_n)} \prod_{k=1}^n \frac{1}{1-\frac{1}{p_k}})=1.781072\dots$$so we can say$$ 1.78 \ ln(p_n) \sim \prod_{k=1}^n \frac{1}{1-\frac{1}{...
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The series at hand is given by, $$\sum_{k=1}^{\infty} \left(\frac{2k+1}{k (k+1)}\right)^s$$ I think it converges for $s>1$, but I have not been able to derive a general expression for this series. ...