# Tagged Questions

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.

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### Evaluate $\displaystyle \sum_{n=1}^{\infty }\int_{0}^{\frac{1}{n}}\frac{\sqrt{x}}{1+x^{2}}\mathrm{d}x.$

I have some trouble in evaluating this series $$\sum_{n=1}^{\infty }\int_{0}^{\frac{1}{n}}\frac{\sqrt{x}}{1+x^{2}}\mathrm{d}x$$ I tried to calculate the integral first, but after that I found the ...
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### Proving that $\sum\limits_{n=2}^{\infty }\frac{\zeta (n)}{2^{n-1}}=\log(4)$

$$\frac{\zeta (2)}{2}+\frac{\zeta (3)}{2^2}+\frac{\zeta (4)}{2^3}+\frac{\zeta (5)}{2^4}+...=\log(4)$$ I tried to prove it, but the problem with the odd zeta terms so that I don't have a function ...
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### Sum of Infinite Series with the Gamma Function

I am computing the volume of an infinite family of polytopes and have run into the following sum, which I am not sure how to evaluate, as it looks similar to the Riemann zeta function, except with the ...
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I am reading Stein and Shakarchi's Fourier Analysis text and the proof Dirichlet's theorem and I am looking for clarification on how he derives the following for large $s$, $\lim_{s\to\infty}$ and $\... 1answer 82 views ### Where are the zeros of$\prod\limits_p (1-(p-1)^z)$? Define$f(z)$as the analytic continuation of$\prod\limits_p (1-(p-1)^z)$where$z$is complex and the product is over the odd primes$p$. Where are the zeros ($f(z)=0$) of this function ? 1answer 199 views ### Zeta-like function with offset Is there a known function of the form: $$\zeta(s,a) = \displaystyle\sum_{n=1}^\infty\frac{1}{n^s+a},$$ and if so what is its name? 2answers 197 views ### Lerch-$\small \zeta(\varphi,0,-n)$of integer *n* purely real and imaginary ($\small \zeta_\varphi (-n)^2 $is real) for$\small n \ge 2$? Are the Lerch-$\zeta(\varphi,0,-n) $of integer n (for shortness I use the notation of my earlier question$\small \zeta_\varphi(-n)$) periodically purely real and imaginary:$\zeta_\varphi (-n)^2 $... 1answer 110 views ### Mysterious Inverse Mellin transform using residue theorem The origin of this problem lies in the explanation of the evaluation of the series$\sum_{n\geq1}\frac{\cos(nx)}{n^2}=\frac{x^2}{4}-\frac{2\pi}{4}+\frac{\pi^2}{6}$see this link ( Series$\sum_{n=1}^...
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The integral \begin{align} I_{4} = \int_{0}^{1} \ln(1-x) \ \ln^{2}\left( \ln\left(\frac{1}{x}\right) \right) \ \frac{dx}{x} \end{align} can be expressed as \begin{align} I_{4} = \zeta^{''}(2) - \frac{\...
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### How to show $\zeta (1+\frac{1}{n})\sim n$

How to show $\zeta (1+\frac{1}{n})\sim n$ as $n\rightarrow \infty$ where $\zeta$ is the Riemann zeta function. And can we say $\lceil \zeta (1+\frac{1}{n}) \rceil=n$ for any positive integer $n\geq 1$...
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### Find: $\zeta \left( 3,1,1,1 \right)$

While solving a summation, I came across this: $$\zeta \left( 3,1,1,1 \right)=?$$ I'm new to multiple zeta values. That's why I couldn't find this. So my question is does a closed form exist ...
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### Proof of Functional Equation Zeta

$$\pi^{-s/2}\zeta(s)\Gamma(s/2)=\pi^{-(1-s)/2}\zeta(1-s)\Gamma((1-s)/2)$$ (That's the equation that want to prove) Hello guys, so I'm trying to prove the functional equation of Riemann Zeta, ...
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### What are all functions of the form $\frac{\cosh(\alpha x)}{\cosh x+c}$ self-reciprocal under Fourier transform?

There are some functions that are self reciprocal under cosine Fourier transform: $$\frac{1}{\cosh x}, \frac{\cosh x}{\cosh 2x},\frac{1}{1+2\cosh x}$$ It seems that they ...
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### Any possible suspects for $\zeta(3)$?

I'm young, and have been studying this number for quite some time. Possible suspects for a closed form i have personally encountered through ghetto makeshift studyies are: Euler-Mascheroni Constant ...
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### Pointwise convergence domain of function series

I'm stuck at finding pointwise convergence domain of the following function series $$\sum_{n=1}^\infty \frac{\sqrt[3]{(n+1)}-\sqrt[3]{n}}{n^x+1}$$ I tried to use d'Alembert and Weierstrass tests, but ...
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### Prove that this summation evaluates out to $\zeta(2)-1$

I am aware of the following identity: $$\sum_{m=1}^\infty \left(\frac{1}{m}-\left(\zeta(2)-\sum_{n=1}^m \frac{1}{n^2}\right)\right)=\zeta(2)-1$$ I can't quite figure out how to prove this result. ...
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### Dirichlet series experiment - computing the rational coefficient

Let consider the sequence of numbers $a_n = 0,1,-1,0,1,-1,0,1,-1, ...$ extended periodically ( so it has period $9$, $a_{n+10}=a_n$. In fact, this is a Dirichlet character $a_n = \chi_9(n)$ modulo 9. ...
I want to find a closed form for the following infinite sum: $$\sum_{k=2}^{\infty} \frac{(-1)^k\cdot(k-1)}{k\cdot(k+1)}\cdot \zeta(k)$$ Is it possible? My approach was to transform it into a double ...
### Infinite Series $\sum_{k=1}^\infty\left(\zeta(2)-\sum_{n=1}^k\frac1{n^2}\right)^2$ [duplicate]
Evaluate: $$\sum_{k=1}^\infty\left(\zeta(2)-\sum_{n=1}^k\frac1{n^2}\right)^2$$ Recognizing that $\zeta(2)-\sum_{n=1}^k\frac1{n^2}$ can be written as $\psi_1(1+k)$ where $\psi_1(z)$ is the trigamma ...