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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14
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4answers
3k 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 ...
0
votes
1answer
19 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} ...
1
vote
1answer
43 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
vote
2answers
44 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
votes
0answers
392 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 ...
2
votes
1answer
96 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 $$ ...
6
votes
1answer
283 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 ...
3
votes
2answers
57 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 $s=0$: ...
7
votes
2answers
229 views

Riemann zeta function and modulus

The functional equation for the zeta function $ζ(s)$ is given by $ζ(s)=f(s)ζ(1-s)$ (a) We know that if $Re(s)=1/2$, then $|f(s)|=1$. My question is about the case where $|f(s)|=1$ outside the ...
4
votes
0answers
125 views

Is the infinite sum $\sum_{s=2}^\infty \frac{\zeta(s)}{s!}$ known? If so, what is its value?

I recently ran into this infinite sum: $$\sum_{s=2}^\infty \frac{\zeta(s)}{s!}$$ and have tried to solve it to no avail. Any references, solutions, or general advice would be greatly appreciated.
7
votes
1answer
105 views

Can derivative of Hurwitz Zeta be expressed in Hurwitz Zeta?

Can the derivative of Hurwitz Zeta function by the first argument be expressed in terms of Hurwitz Zeta and elementary fuctions? There is a formula which expresses Hurwitz Zeta through its ...
3
votes
0answers
400 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} ...
5
votes
1answer
79 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} ...
5
votes
0answers
140 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 ...
3
votes
1answer
77 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 ...
1
vote
1answer
103 views

Can the Riemann Zeta derivative be expressed in terms of Riemann Zeta?

From this question: http://mathoverflow.net/questions/134600/zetax-in-terms-of-zetax-zeta1-x-gamma-psi it seems that Zeta can be expressed through its derivative: $$\zeta(1-x) = ...
0
votes
1answer
46 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 ...
5
votes
0answers
166 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 ...
0
votes
0answers
36 views

Provide me notes on Riemann zeta function to boast my knowledge to use in Research on Analytical Number Theory

I need your help. I want to study the Riemann zeta function from the very basic level, its concepts, theorems, solved problems etc. I am assigned one problem from Analytical Number Theory related to ...
0
votes
0answers
24 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?
5
votes
0answers
71 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 = ...
5
votes
0answers
64 views

Why does the tribonacci constant have a trilogarithm ladder?

When I came across the dilogarithm ladders of Coxeter and Landen, namely, $$\text{Li}_2(\alpha^6)-4\text{Li}_2(\alpha^3)-3\text{Li}_2(\alpha^2)+6\text{Li}_2(\alpha)-\tfrac{7}{5}\zeta(2)=0\tag1$$ ...
3
votes
1answer
99 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 ...
2
votes
1answer
27 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
vote
4answers
89 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 ...
0
votes
0answers
17 views

generating function for the Hurwitz Zeta

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)+ ...
0
votes
1answer
18 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}) ...
2
votes
0answers
47 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 \ | \ ...
3
votes
0answers
135 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 ...
1
vote
0answers
36 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} ...
44
votes
0answers
792 views

Gross-Zagier formulae outside of number theory

(Edit: I have asked this question on MO.) The Gross-Zagier formula and various variations of it form the starting point in most of the existing results towards the Birch and Swinnerton-Dyer ...
8
votes
1answer
130 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 ...
1
vote
1answer
33 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 ...
0
votes
1answer
24 views

Generating function for the zeta function

Given a Hamiltonian $H$, with a spectrum of eigenvalues $\lambda$, you can define its zeta function as $\zeta_H(s) = tr \frac{1}{H^s} = \sum_{\lambda}^{} \frac{1}{\lambda^s}$. Subsequently, the log ...
2
votes
0answers
55 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 ...
3
votes
1answer
73 views

Moving the integral $Q(x) = -\frac{e^{-1/2x}}{4i}\int_{1/2-i\infty}^{1/2+i\infty} \zeta(s)\Gamma(\frac{s}{2})\pi^{-s/2}e^{xs} ds$ past Re(s) = 1.

Given the integral $$Q(x) = -\frac{e^{-1/2x}}{4i}\int_{1/2-i\infty}^{1/2+i\infty} \zeta(s)\Gamma(\frac{s}{2})\pi^{-s/2}e^{xs} ds,$$ I know that the integrand is holomorphic except for simple poles at ...
1
vote
0answers
32 views

Generalized Hurwitz Zeta function

Let $d\ge 1$ be an integer, $a>0$ be a real number and let $\vec{s} := (s_0,\cdots,s_{d-1})$ where all the components are strictly bigger than one. We generalize the zeta function to higher ...
4
votes
1answer
96 views

Proving that $\sum_{n=2}^{\infty }\frac{\zeta (n)}{2^{n-1}}=\log(4)$

$$\frac{\zeta (2)}{2}+\frac{\zeta (3)}{2^2}+\frac{\zeta (4)}{2^3}+\frac{\zeta (5)}{2^4}+...=\log(4)$$ I tried to prove it, but the problem with the odd zeta terms so that I don't have a function ...
3
votes
5answers
307 views

How are Zeta function values calculated from within the Critical Strip?

We note that for $Re(s) > 1$ $$ \zeta(s) = \sum_{i=1}^{\infty}\frac{1}{i^s} $$ Furthermore $$\zeta(s) = 2^s \pi^{s-1} \sin \left(\frac{\pi s}{2} \right) \Gamma(1-s) \zeta(1-s)$$ Allows us to ...
3
votes
2answers
131 views

Can I get a closed-form of $\frac{\zeta(2) }{2}-\frac{\zeta (4)}{2^3}+\frac{\zeta (6)}{2^5}-\frac{\zeta (8)}{2^7}+\cdots$?

Can I get a closed-form of $$\frac{\zeta(2) }{2}-\frac{\zeta (4)}{2^3}+\frac{\zeta (6)}{2^5}-\frac{\zeta (8)}{2^7}+\cdots$$
8
votes
3answers
142 views

How can prove that $\sum_{n=1}^{\infty }\frac{\zeta (2n)}{4^{n-1}}(1-\frac{1}{4^n})=\frac{\pi }{2}$

$$\zeta (2)(1-\frac{1}{4})+\frac{\zeta (4)}{4}(1-\frac{1}{4^2})+\frac{\zeta (6)}{4^2}(1-\frac{1}{4^3})+...=\frac{\pi }{2}$$ The WolframAlph couldn't recognize the closed-form which is $\pi/2$ ...
31
votes
3answers
823 views

How to prove that $\frac{\zeta(2) }{2}+\frac{\zeta (4)}{2^3}+\frac{\zeta (6)}{2^5}+\frac{\zeta (8)}{2^7}+\cdots=1$?

How can one prove this identity? $$\frac{\zeta(2) }{2}+\frac{\zeta (4)}{2^3}+\frac{\zeta (6)}{2^5}+\frac{\zeta (8)}{2^7}+\cdots=1$$ There is a formula for $\zeta$ values at even integers, but ...
40
votes
0answers
1k views

The log gamma integral $\int_{0}^{z} \log \Gamma (x) \ \mathrm dx$

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) + ...
3
votes
1answer
136 views

Extending the zeta function to semiprimes, etc.

The Riemann Zeta function is defined for $s > 1$ as \begin{align} &\prod _{n=1}^{\infty}\dfrac{1}{1 -\ p_{n}^{\ \ -s}}\\ \end{align} It is possible to extend the zeta function to semiprimes ...
0
votes
1answer
54 views

Formula for tangent derivatives, how to prove?

How to prove? $$(\tan x)^{(s-1)}=\pi^{-s}\Gamma(s)\left(\zeta\left(s, \frac12-\frac x\pi\right)+(-1)^s\zeta\left(s, \frac12+\frac x\pi\right)\right) $$
0
votes
0answers
37 views
0
votes
0answers
29 views

Can you provide a lower bound on $|\zeta(\sigma+it)|$ for $\sigma$<0?

To prove the theorem 9.12 of "The theory of the Riemann Zeta-Function", I have to show that $|\zeta(\sigma+it)|$ > $A|t|^c$ for $\sigma$<0 and sufficiently large t. How can I show this?
1
vote
1answer
36 views

Stirling formula on $\frac{\gamma(\frac{1}{2}s)}{\gamma(\frac{1-s}{2})} = O(|t|^{\sigma - \frac{1}{2}})$

I tried to prove $\frac{\Gamma(\frac{1}{2}s)}{\Gamma(\frac{1-s}{2})} = O(|t|^{\sigma - \frac{1}{2}})$ using stirling formula when $s =\sigma+it$. However, since stirling formular for gamma function is ...
1
vote
1answer
99 views

Galois theory on curves

Context: Let $\mathbb{F}$ be the algebraic closure of $\mathbb{F}_q$ for $q$ prime. We know that $\mathbb{F}(t)$ for $t$ transcendental is the function field of the projective line ...
4
votes
1answer
34 views

coefficients of the zeta function of curve over a finite field $\mathbb{F}_q$

Let $C$ be a non-singular curve over $\mathbb{F}_q$. Denote by $d$ the degree map from the group of divisors to $\mathbb{Z}$ and denote by $P$ the set of prime divisors w.r.t. to the function field. ...