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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43 views

A shortcut for analytic continuation?

Let $P(x)$ be a nonconstant integer polynomial with nonnegative coëfficiënts such that the equation $y= P(y)$ has only one real solution $q$. Let $x_1=P(0)$ and $x_n = P(x_{n-1})$. $$f(z) = \sum_{n&...
1
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1answer
33 views

Breaking up integral representations by convergence

A known integral takes the form of $$\zeta(3)=\frac{1}{2}\int_{0}^{\infty} \frac{t^2}{e^t-1}dt$$ Through Wolfram part of the integral converges to $$\int_{0}^{\infty} \frac{t}{e^t-1}dt = \frac{\pi^...
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0answers
40 views

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 $$\frac{1}{\pi{i}}\Bigl(G(z)-2G(2z)\Bigr) = -F(z)+F(-z)$$ ...
11
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2answers
352 views

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 ...
0
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0answers
21 views

zeta function variety

I'm trying to understand the motivation for zeta function of a variety over a finite field, that is the connection of the standard definition $$ \exp\left( \sum_{n=1}^\infty \frac{N_n t^n}{n} \right) ...
3
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2answers
64 views

Clausen zeta function

For $0 < \theta < 2\pi$, define $$\kappa(x,\theta) = \frac{1}{\zeta(x)}\sum_{n=1}^\infty \frac{e^{ in\theta}}{n^x}$$ for $\Re(x) > 1$. It is easy to see that $$\kappa(x,\theta) = \frac{1}{\...
4
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2answers
437 views

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: \begin{equation} \frac{1}{\cosh x}, \frac{\cosh x}{\cosh 2x},\frac{1}{1+2\cosh x} \end{equation} It seems that they ...
6
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3answers
117 views

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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0answers
18 views

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 ...
7
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1answer
90 views

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 $\zeta(2)$, ...
1
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1answer
49 views

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
2
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1answer
113 views

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 ...
8
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2answers
230 views

Is there a closed-form of $\frac{\zeta (2)}{\pi ^2}+\frac{\zeta (4)}{\pi ^4}+\frac{\zeta (6)}{\pi ^6}+…$

The value of $$\frac{\zeta (2)}{\pi ^2}-\frac{\zeta (4)}{\pi ^4}+\frac{\zeta (6)}{\pi ^6}-.....=\frac{1}{e^2-1}$$ Is there a closed-form of $$\frac{\zeta (2)}{\pi ^2}+\frac{\zeta (4)}{\pi ^4}+\frac{\...
3
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3answers
106 views

Why does the $\sum_{n>1}(\zeta(n)-1)=1?$

While I was looking at the values of the zeta function for the first natural numbers, I noticed that the sum of the values minus $1$, converge to $1$. Better put: $$\sum_{n=2}^{\infty} \zeta(n)-1 = 1 $...
3
votes
3answers
190 views

Prove that $\zeta(-1)=\zeta(-13)$.

Basically what the title says. I saw this through another Math Platform but did not get any response to it. The original question was to find distinct integers of $ x $ and $y$ such that $ \zeta(x) =...
2
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0answers
211 views

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 ...
3
votes
1answer
36 views

A certain zeta function; or, the determinant of the Laplacian plus a constant on the circle

I am interested in a certain "zeta function," a meromorphic function of $s \in \mathbb{C}$ that depends on a real parameter $\alpha \neq 0$. It's defined for the real part of $s$ large by $$ \zeta_\...
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2answers
122 views

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, ...
1
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1answer
83 views

Closed form of this sum

$$\sum _{ s=1 }^{ \infty }{ \left( \frac { 1 }{ 4s-1 } \sum _{ n=0 }^{ \infty }{ \left( \frac { 1 }{ n+1 } \sum _{ k=0 }^{ n }{ \left( \left( \begin{matrix} n \\ k \end{matrix} \right) \frac { { \left(...
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3answers
111 views

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 ...
5
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1answer
78 views

Is the relation $P(n) \sim \frac{1}{2^n}$ already known?

Apologies in advance if there is a violation of rules/laws here, as I am not a mathematician. $$ \begin{align} \lim_{n\to\infty} \left( \frac{\pi^{n}}{\zeta(n)}P(n) \right)^{\frac{1}{n}} &= \...
3
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1answer
79 views

Calulate a limit involving $\zeta{(\zeta{(z)})}$

I'm currently trying to evaluate the following limit: $$ \lambda=\lim_{z\to\infty}{\left[2^z-\left(\frac{4}{3}\right)^z-\zeta{(\zeta{(z)})}\right]} $$ A look at numerical approximations suggests, that ...
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0answers
32 views

Increasing sequences and $\zeta$-type functions

The Riemann zeta function is defined as the sum $\zeta(s) = \sum_{n \geqslant 0} n^s$. The question is whether it globally characterizes the sequence of all natural numbers, in the following sense: ...
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0answers
50 views

Hasse-Weil zeta function of projective hypersurfaces

Assume $f$ is a homogeneous integer polynomial in $n\geq 3$ variables such that the hypersurface $f=0$ is irreducible over $\mathbb{Q}$ (but not necessarily over $\overline{\mathbb{Q}}$ so for example ...
6
votes
0answers
84 views

Asymptotic behavior of the generalized polygamma function

The generalized polygamma function$^{[1]}$$\!^{[2]}$ is defined as $$\psi^{(\nu)}(z)=e^{-\gamma\!\;\nu}\;\partial_\nu\!\left(\frac{e^{\gamma\!\;\nu}\;\zeta(\nu+1,z)}{\Gamma(-\nu)}\right),\tag1$$ where ...
1
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1answer
66 views

Eulers proof sum of natural numbers

I've to recheck Eulers proof of the sum of the natural numbers, but I dont now exactly what it is? It has something to do with the $\zeta(s)$? Thanks in advance
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2answers
72 views

What is sum: $\sum\limits_{m,n\geq1}\frac{1}{(1+mn)^2}$?

What is the sum $$\sum\limits_{m,n\geq1}\frac{1}{(1+mn)^2}.$$
0
votes
1answer
22 views

Differentiating the spectral zeta function

I need to figure out how $\sum_{k=1}^{n} \text{ln}\ \lambda_{k} = - \frac{d}{ds}\Big|_{s=0} \sum_{k=1}^{n} \lambda_{k}^{-s}$. But, if I evaluate $- \frac{d}{ds}\Big|_{s=0} \sum_{k=1}^{n} \lambda_{k}^{...
1
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1answer
63 views

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 ...
1
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2answers
48 views

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.
3
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2answers
73 views

Regularizing the $\log\log n$ series

The divergent series $$\sum_{n=1}^\infty\log n$$ can be regularized using the derivative of the Riemann zeta function at $s=0$: $$-\frac{\mathrm{d}}{\mathrm{d}s}\zeta(s)=-\frac{\mathrm{d}}{\mathrm{d}...
7
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2answers
472 views

What is the Möbius analoge for Ihara's $\zeta$ function?

The Dirichlet series that generates the Möbius function is the (multiplicative) inverse of the Riemann zeta function; if s is a complex number with real part larger than 1 we have $$\sum_{n=1}^\...
5
votes
1answer
103 views

Cauchy-Ramanujan Formula $ \displaystyle \sum_{\stackrel{m \in \mathbb{Z}}{m \neq 0}} \frac{\coth m \pi}{m^{4p+3}} $

Cauchy and Ramanujan both gave the formula: $$ \sum_{\stackrel{m \in \mathbb{Z}}{m \neq 0}} \frac{\coth m \pi}{m^{4p+3}} = (2\pi)^{4p+3}\sum_{k=0}^{2p+2} (-1)^{k+1} \frac{B_{2k}}{(2k)!}\frac{B_{4(...
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0answers
160 views

Can these integrals be represented in closed form?

This paper in the formula F.3.6 (page 271) gives the following formula for the derivative of Hurwitz Zeta function: $$\frac d{dz}\zeta(z,a)=-\frac12\ln(a)a^{-z}-\frac{\ln(a)a^{1-z}}{(z-1)^2}-\int_0^\...
3
votes
1answer
103 views

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 ...
0
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1answer
57 views

Does this summation (involving binomial) have a closed form? If so, what is it?

The following sums are the ones I'm interested in: $\sum_{i=m}^{\Omega}{\binom{i}{m}i^{-k}}$ $\lim_{\Omega\rightarrow\infty}{\sum_{i=m}^{\Omega}{\binom{i}{m}i^{-k}}}$ I already know that $\lim_{\...
5
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0answers
180 views

Summation of a function 2

Let $n$ is a positive integer. $n = p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}$ is the complete prime factorization of $n$. Let me define a function $f(n)$ $f(n) = p_1^{c_1}p_2^{c_2}\cdots p_k^{c_k}$ where ...
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0answers
28 views

Kronecker Zeta function

If we define the Kronecker symbol K(a,n) as at Wikipedia, can we define $$\zeta_K(s,a) = \sum_n \dfrac{K(a,n)}{n^s}$$? If so, what does it equal?
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1answer
99 views

Summation of a function

Let $n$ is a positive integer. $n = p_1^{e_1}p_2^{e_2}...p_k^{e_k}$ is the complete prime factorization of $n$. Let me define a function $f(n)$ $f(n) = p_1^{c_1}p_2^{c_2}...p_k^{c_k}$ where $c_k = ...
2
votes
1answer
36 views

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 ...
1
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4answers
103 views

$\sum\limits _{n=1}^{\infty}\frac{1}{n^s}=50$ Riemann-Zeta

I've to find a value for 's' were the infinit sum gives me the value 50. Is that possible and how do I've to calculate that value. I've no idea how te begin so, help me! Solve s for: $$\sum\limits ...
2
votes
1answer
58 views

Generating function for the Hurwitz Zeta: $\sum_{k=1}^{\infty}\zeta_{H}(k+1,a)(-z)^{k}$

We know from the digamma function $$ \Psi (z+1)= -\gamma -\sum_{k=1}^{\infty}\zeta(k+1)(-z)^{k} $$ My question is if there is a similar formula for $$ f(a)+ \sum_{k=1}^{\infty}\zeta_{H}(k+1,a)(-z)^...
4
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1answer
134 views

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 ...
0
votes
1answer
24 views

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{...
2
votes
0answers
54 views

zeta function of abelian varieties and the exterior algebra

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_{...
3
votes
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 ...
1
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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)...
3
votes
0answers
150 views

Why the equality of spectral zeta functions imply the isospectrality?

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 ...
8
votes
1answer
156 views

Proving $\zeta(2) - \beta(1) + \zeta(4) - \beta(3) + \zeta(6)- \beta(5) + \ldots=1$

Trying to prove $$\zeta(2) - \beta(1) + \zeta(4) - \beta(3) + \zeta(6)- \beta(5) + \ldots=1$$ I found by numerical calculation that (when $k$ goes to infinity) $$\sum_{n=1}^{k}\zeta (2n)=k+3/4+o(1),$...
2
votes
1answer
98 views

Convergence of the series for the Dedekind zeta function

The Dedekind zeta function of an algebraic number field $K$ is defined as $\zeta_K(s)\mathrel{\stackrel{\rm def}=} \sum\limits_{I \subset\mathcal O_K} \frac{1}{(N_{K/\mathbb Q}(I))^s}$, where $N_{K/\...