# 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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### Problem inside the derivation of $\zeta(-k)= -\frac{B_{k+1}}{k+1}$

I am having trouble with one step for the derivation of $\zeta(-k)= -\frac{B_{k+1}}{k+1}$ found here. In the below steps, how do we get from $$\frac{1}{\pi{i}}\Bigl(G(z)-2G(2z)\Bigr) = -F(z)+F(-z)$$ ...
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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 ...
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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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### Is $\sum_{n=1}^{\infty }\frac{8n\cdot\zeta (2n)}{3\cdot 2^{2n}}=\zeta (2)$?

$$\sum_{n=1}^{\infty }\frac{8n\cdot\zeta (2n)}{3\cdot 2^{2n}}=\zeta(2)$$ By using numerical calculation, I found this relationship between the values of zeta function at even integers and $\zeta(2)$,...
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### Indicating functions as series on posets

Let $\mathcal{P}$ be a poset, which we can take to be finite for simplicity, and $\zeta$ its zeta function. Define $\alpha:=1-\zeta^{-1}$, i.e. $\zeta=1/(1-\alpha)$. Does there exist a formal series ...
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### Proving that $\sum_{n=0}^{\infty }\frac{3(n!)^2}{(2n+2)!}=\sum_{n=1}^{\infty }\frac{1}{n^2}=\frac{\pi ^2}{6}$

Proving that $$\sum_{n=0}^{\infty }\frac{3(n!)^2}{(2n+2)!}=\sum_{n=1}^{\infty }\frac{1}{n^2}=\frac{\pi ^2}{6}$$ I know the proving of second series which is very famous series to give us $\zeta(2)$, ...
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### finding the closed-form of $k$ in the series $\sum_{n=1}^{m}\beta (2n-1)\zeta (2n)=m+k$

finding the closed-form of $k$ in the series $$\sum_{n=1}^{m}\beta (2n-1)\zeta (2n)=m+k$$ when m go to infinity from some values of $m$, I found the $$k=0.358971008185307705...$$ any help, thanks
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### The values of the derivative of the Riemann zeta function at negative odd integers

I would like to know if the values of the derivative of the Riemann zeta function at negative odd integers are computed, i.e. $\zeta'(-n)$ when $n$ is odd. When I look at the page from Wolfram ...
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### Abel-Plana formula for $\zeta(s)$, is this integral approximation correct?

I wrote a computer program to calculate values for $\zeta(s)$. I was scanning for something that would calculate complex values for $\zeta(s)$. I found the following approximation under the Integral ...
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### Sum involving zeta functions

Find closed form of the following - $$\displaystyle \sum_{n=2}^{\infty}{\left(\frac{(n-1)\zeta(n)}{4n-1}\right)}$$ I don't know how to approach to it - Using the integral definition? I cannot use ...
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### Growth of the zeta function on the line $Re(s)=\frac{1}{2}$

I've seen that on the line $Re(s)=\frac{1}{2}$, $\zeta(s)=O(t^{\frac{1}{4}})$ where, as usual, $s=\sigma+it$. My teacher has told me that this can be derived directly from the functional equation of ...
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### How do I evaluate this $\lim \frac{\zeta(n)} {({n)!}} , n\to\infty$ if it was existed?

Is there someone who can show me how do I evaluate this limit $$\lim_{n\to +\infty} \frac{\zeta(n)} {n!}$$ if it exists ? Thank you for any help.
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### Concerning Hurwitz Zeta function, how to prove the following identity?

It is claimed that $$\zeta'(0,s)=\ln\left(\frac{\Gamma(s)}{\sqrt{2\pi}}\right)$$ where the derivative is meant by the first argument (as usual with Hurwitz Zeta). How to prove this? Wolfram Alpha ...
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### Zeta Function : Identify This Variant of an $L$- Series

This is my first question on Math.SE ; if I am wrong somewhere , please correct me I believe that there are not any mentioned variations of a Zeta function of this form : Where, $w$ is a ...
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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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### zeta-function regularization $\prod_{m,n\in \mathbf{Z}}(m+n\tau+u)=iy^{\frac{1}{2}} q^{\frac{1}{12}} \prod_{n=1}^{\infty} (1-yq^n)(1-y^{-1}q^{n-1})$

I'd like to prove(explicilty compute) the following relation \begin{align} \prod_{m,n\in \mathbf{Z}}(m+n\tau+u)=iy^{\frac{1}{2}} q^{\frac{1}{12}} \prod_{n=1}^{\infty} (1-yq^n)(1-y^{-1}q^{n-1}) \end{...
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Let $X$ be a smooth projective variety over a finite field $k$. By Grothendieck, the zeta function of $X$ admits the cohomological expression $$Z(X, t)=\prod_{j=0}^{2\dim X} \det (1-F t \ | \ H^j(X_{... 0answers 396 views ### What special role plays the function \pi^{\frac x\pi} in analysis? I have tried to redefine some special functions in the most "natural" way, that is the way which allows to simplify the relations the most. I would call these functions "parelementary". The ... 0answers 41 views ### Divergence Dedekind zeta function Let K be a number field, \mathcal O_K be its ring of integers, T a positive integer and N the norm function. Give an upper bound (in T) for$$\sum_{I\leq \mathcal O_K: N(I)\leq T} \frac{1}{N(I)...
Let $\Delta_{M_1}$ and $\Delta_{M_2}$ be the Laplace-Beltrami operators on two compact and connected Riemannian manifolds $M_1$ and $M_2$ respectively. We define the spectral zeta function (or ...