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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Important numerator and denominators in the evaluation of the integral: $\int_0^\infty x^t \operatorname{csch} x\text{ d}x$

$$\int_0^\infty x^t\operatorname{csch}x\text{ d}x=\frac{a\zeta(t+1)}{b}$$ for $t\in\Bbb{N}$ How might one represent $a,b$ in terms of $t$? (Note that $a,b\in \Bbb{N}$) If possible, could one also ...
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45 views

Why was the zeta function introduced?

I know the 'Zeta Function' is very useful in Mathematics, and that it has relations with many other functions (such as the 'Gamma Function'). I also know the 'Zeta Function' $\zeta(s)$ is defined as: ...
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0answers
28 views

Euler Product Formula - Zeta Function

For $s \in \mathbb{C}$ and $\sigma = \Re(s)>1$, $$\zeta(s) = \prod\limits_{p \in \mathbb{P}}\left(1 - \frac{1}{p^s}\right)^{-1}$$ My question is: is the above correct? Or should the $s$ be ...
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0answers
63 views

Assigning values to a divergent integral?

Question If I can assign the series of the zeta function to: $$ \zeta(-1) \to 1+2+3+\dots$$ why can't we assign the integral $$ \int_{0}^{\infty} x dx \to 0$$ and it still have some physical ...
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1answer
2k views

Evaluating the log gamma integral $\int_{0}^{z} \log \Gamma (x) \, \mathrm dx$ in terms of the Hurwitz zeta function

One way to evaluate $ \displaystyle\int_{0}^{z} \log \Gamma(x) \, \mathrm dx $ is in terms of the Barnes G-function. $$ \int_{0}^{z} \log \Gamma(x) \, \mathrm dx = \frac{z}{2} \log (2 \pi) + ...
7
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1answer
182 views

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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0answers
32 views

Can someone expain to me what's going on (binomial coefficient)?

I'm watching this proof for $\zeta(2n)$ on YouTube. This is what I can understand so far: $${s\over e^{s} -1} = \sum^{\infty}_{n=0} {\beta_n\over n!} s^n$$ Where $\sum^{\infty}_{n=0} {\beta_n\over ...
7
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2answers
421 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 $$ ...
2
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2answers
169 views

Where do these infinites Tao is talking about come from?

I was reading about how 1+2+3+4... !=-1/12 (which is something that drove me crazy when I first heard about it in a Numberphile video) in an article by Terence Tao. He says that -1/12 is in fact -1/12 ...
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1answer
20 views

Bounds for zeta function residue

Let $K$ be an algebraic number field and let $c = c(K)$ denote the residue at $s = 1$ of its zeta function. It is known Wikipedia: class number formula that c can be determined via $$c = \frac{2^r ...
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2answers
59 views

Convergence of prime zeta function for $\mathfrak R(s)=1$?

By doing some estimates for the partial sums of the Prime zeta function $P(s)=\sum_p p^{-s}$ for $\mathfrak R(s)=1$ I got that $P(1+i\alpha)$ converges for every $\alpha\neq0$... Since I did not ...
3
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0answers
138 views

Riemann vs. Ihara's $\zeta$ Function Variable Question

The Euler product for the Riemann zeta function $\zeta(z)$ implies that $$ \log\zeta_R(z)=\sum_{m>0}\frac{P(mz)}{m} \tag{R}, $$ whereas the Ihara zeta function for a graph $G$--all can be ...
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1answer
9 views

What is the meaning of cyclically equivalence classes of multiple indices?

Can you please give me an example for the cyclic equivalence classes of multiple indices on the following set? $$ I(w,d)=\{\ (k_1,...k_d)\mid k_1+\ldots+k_d=w, k_1,...k_d \ge 1\,\}$$ where $ w$: ...
2
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1answer
70 views

A limit involving the Hurwitz zeta function

I want to show that $$ \lim_{s \to 1} \left( \zeta(s,a) - \frac{1}{s-1} \right) = - \psi(a)$$ where $\zeta(s,a)$ is the Hurwitz zeta function and $\psi(a)$ is the digamma function. The only ...
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1answer
28 views

Seeking possibility of more elementary means of evaluating an improper integral.

It can be shown that $\int_0^\infty -\log{(1-e^{-x})}=\zeta(2)$ by expanding out the integral as $\log(1-z)$, exchanging summation and integration, then summing up the integrals. I am wondering if ...
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1answer
267 views

All Ihara $\zeta$ functions for planar $k$-regular graphs with a given set of faces are equivalent

This sounds like a simple piece of math (which got a long story over time, thanks for reading!) and the consequence seems surprising. At least to me. Here it is: It boils down to comparing two ...
3
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2answers
60 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) = ...
3
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1answer
67 views

Is there a closed form for the product of odd zetas?

$$\prod_{n=1}^\infty \zeta(2n+1)=\zeta(3)\zeta(5)\cdots$$ I have only managed to prove that this converges due to comparison with Euler's formula for $\zeta(2n)$ Is there a closed form for that ...
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1answer
47 views

How do I evaluate this sum :$ \sum_{n=1}^{\infty}\frac{{(-1)}^{n-1}\log n}{n^s}$?

How do I evaluate this sum :$$ \sum_{n=1}^{\infty}\frac{{(-1)}^{n-1}\log n}{n^s}$$ Note : In wolfram alpha it is convergent for $Re(s)>1$ .!! Thank you for any help
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183 views

A map from zeros of $\zeta(s)$ to zeros of $C(s)?$

Let $P(s),C(s),\zeta(s)$ be the prime zeta function, the analogous composite zeta function, and the classical zeta function. I do not know whether it is known that there are infinitely many zeros of ...
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1answer
49 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)+ ...
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1answer
35 views

Books on Zeta Regularization Product

Does anybody know some book on zeta regularization, and the zeta regularization product? I'm quite interested on the topic but I would need a book with some review...
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1answer
35 views

Does the Weil conjecture work with 0 dimensional varieties?

Suppose I have some inseparable quadratic polynomial $p(x)$ over $\mathbb{F}_q$ which has a pair of roots in $\mathbb{F}_{q^2}$. If I compute the zeta function of the variety cut out by $p(x)$ (in ...
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0answers
26 views

Proof that $\zeta_P$ never does the following…

Let's assume that $\zeta_P(s)$ is the prime-zeta function or: $$\zeta_P(s)=\sum_{n\in P} \frac{1}{n^s}$$ I noted that if $\forall s\in \Bbb{Q},s\not =0, \zeta_P (s)\not\in \Bbb{Q}$ I cant really ...
2
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0answers
94 views

Zeros of the prime zeta function

A basic confusion about zeros of the prime zeta function $P(s).$ Let $s= \sigma+i~t$ with $\sigma>0.$ Letting $C(s)$ be the corresponding composite zeta function we have ...
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0answers
28 views

Zeta function of $y^2 = x^3 - x$ over Fp

Zeta function of $y^2 = x^3 - x$ over Fp, where p = 3(mod 4) Can someone give an explanation of a zeta function? I've tried researching it, and I cannot seem to understand. Is there some kind of ...
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0answers
4 views

does the $\zeta_K$ function of a function field determine the genus of that function field?

Let $K_1$ and $K_2$ both be function fields over a finite field (or algebraic curves, if you like) with zeta functions $\zeta_{K_1}$ and $\zeta_{K_2}$. Say that $\zeta_{K_1} = \zeta_{K_2}$ - so that ...
4
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0answers
113 views

Finding zeta function of an elliptic curve

Let p=3 (mod 4) be a prime, and $E/F_{p^r}$ be the elliptic curve given by $y^2 = x^3 − x$ Find the zeta-function of $E/F_p$ and use it to determine $|E(F_{p^r} )|$ for all r>0.
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3answers
209 views

Zeta function for negative integers

I already proved that $\zeta(z)=\frac{1}{\Gamma(z)}\int_0^\infty\frac{t^{z-1}}{e^t-1}dt=\frac{\Gamma(z-1)}{2\pi i}\int_{-\infty}^0\frac{t^{z-1}}{e^{-t}-1}dt$ Now the Benoulli numbers are defined by ...
4
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1answer
66 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 ...
3
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2answers
47 views

Closed form of the series $\sum\frac{\ln(n)}{n^a}$

I want to know if there exists, and how to arrive at, a closed form of this infinite sum: $$S_a=\sum_{n=1}^\infty \frac{\ln(n)}{n^a}$$ I know the series converges at least for every $a>1$ by the ...
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0answers
36 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) = ...
6
votes
2answers
409 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 ...
1
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1answer
27 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 = ...
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3answers
91 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 ...
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2answers
258 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 ...
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1answer
418 views

How to get from Chebyshev to Ihara?

I have competing answers on my question about "Returning Paths on Cubic Graphs Without Backtracking". Assuming Chris is right the following should work. Up to one thing: The number of returning paths ...
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0answers
35 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 $$\frac{1}{\pi{i}}\Bigl(G(z)-2G(2z)\Bigr) = -F(z)+F(-z)$$ ...
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0answers
19 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) ...
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1answer
124 views

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 ...
15
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4answers
4k views

Factorial of infinity

So, I've read in this article that: $$\zeta'(0) = \log\sqrt\frac{1}{2\pi}$$ And that: $$e^{-\zeta'(0)} = 1\cdot2\cdot3\cdot\ldots\cdot\infty = \infty! = \sqrt{2\pi}$$ I found this result very ...
6
votes
3answers
113 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 ...
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0answers
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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 ...
1
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1answer
45 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
7
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1answer
86 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 $\zeta(2)$, ...
2
votes
1answer
69 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
votes
2answers
214 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 ...
18
votes
3answers
847 views

A Geometric Proof of $\zeta(2)=\frac{\pi^2}6$? (and other integer inputs for the Zeta)

Is there a known geometric proof for this famous problem? $$\zeta(2)=\sum_{n=1}^\infty n^{-2}=\frac16\pi^2$$ Moreover we can consider possibilities of geometric proofs of the following identity for ...
4
votes
2answers
103 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, ...
9
votes
0answers
467 views

An Expression for $\log\zeta(ns)$ derived from the Limit of the truncated Prime $\zeta$ Function

I think, here, I found $$ P_\color{red}x(\color{blue}s)=\sum_{p<\color{red}x} \frac{1}{p^{\color{blue}s}} =\sum_{\color{green}n=1}^{\infty}\frac{ \mu (\color{green}n)}{\color{green}n} ...