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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Riemann Zeta formula

can anyone check if this formula is plausible ?? $$ \frac{1}{\zeta (s)} = \sum_{n=0}^{\infty}\frac{ (-\pi)^{n}(s-1)s}{2n!(s+2n)(s+2n+1)} $$ according to the authors this formula would be valid only ...
2
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3answers
104 views

$\zeta(2 + it) = \zeta(2-it)$

Let $\zeta(s)$ denote the Riemann zeta-function. Show that $\zeta(2 + it) = \zeta(2-it)$ for all real t. Give some hints how to do this one.Thanks in advance.
2
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2answers
190 views

Proving identity $\displaystyle\sum_{j\geq 1}[(j+t)^{-1}-j^{-1}]=\displaystyle\sum_{k\geq 1}\zeta (k+1)(-t)^{k}$

Motivation: In S.J. Patterson's An introduction to the theory of the Riemann Zeta-Function it is proved (p.132) that $\displaystyle -\Gamma ^{\prime }(t)/\Gamma (t)=\gamma +t^{-1}+\underset{j\geq ...
2
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1answer
309 views

Closed-form expressions for $\sum^N_{n}\frac{1}{n^2}$ ($n$, even or odd)

I am trying to find a closed-form expression for the finite sum $$\sum^N_{n=1}\frac{1}{n^2}$$ when $n$ is even and when $n$ is odd. I know that for $N\to\infty$ these series converge towards ...
2
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3answers
116 views

Hint on a limit that involves the Hurwitz Zeta function

I will be honest. Some play with a weird integral has gotten me to this formulation: $$\lim\limits_{n\mathop\to\infty}\frac{\zeta(2,n)}{\frac 1n+\frac 1{2n^2}}=1$$ It seems true because of the ...
2
votes
1answer
149 views

Recurrence relation in complex domain

I am bumping in the following problem : the expansion of $$x^{i/k} \operatorname{LerchPhi}[x,1,i/k]$$ leads, for each value of i, to a linear combination of k terms, each of them writing $$a(j) ...
2
votes
1answer
207 views

What is this series called and when does it diverge?

What is this series called (if it has a name)? When does it diverge without analytic continuation and when does it diverge with analytic continuation? $\sum_{k_1,\dots,k_m=1}^{\infty} ...
2
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1answer
105 views

GCD function and relation between Hurwitz and Riemann zeta function

Does anyone know how to show the following: $\sum_{k=1}^n\gcd(n,k)\zeta(s,\frac{k}{n})=\left(\sum_{k=1}^n\gcd(n,k)\right)\zeta(s)$
2
votes
1answer
109 views

Elementary method to compute $\zeta (3)$

What is an elementary method to compute $\zeta (3)$ which can be understood by a high school student? I know how to compute $\zeta (2)$ but not $\zeta (3)$. Any help is very much appreciated.
2
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2answers
172 views

Estimating the integrated Tchebychev function and calculating its error

I would like to understand how to derive (2) from (1) below. Problem: If $\psi_1$ is the integrated Tchebychev function below $$\psi_1(x)=\frac{1}{2\pi i} ...
2
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1answer
283 views

Periodic Zeta Function Functional Equation

Recall that the periodic zeta function has the Dirichlet series $$F(\lambda,s)= \sum_{n=1}^\infty \frac{e^{2\pi i n\lambda}}{n^s}.$$ This defines an analytic function for $\Re s>0$ and has a ...
2
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1answer
170 views

Show that $f$ is harmonic

Let us consider the function: $$ f(α,β) \equiv \sum_{n = 1}^{\infty}\left(-1\right)^{n - 1}\left[% {n^{2\alpha - 1} - 1 \over n^{\alpha}}\,\cos\left(\beta\ln\left(n\right)\right) \right] $$ My ...
2
votes
1answer
93 views

The multiplication formula for the Hurwitz/generalized Riemann zeta function

I'm having a difficult time showing that $$ \displaystyle \zeta(s,mz) = \frac{1}{m^{s}} \sum_{k=0}^{m-1} \zeta \left(s,z+\frac{k}{m} \right) $$ A couple of authors referred to it as an obvious fact. ...
2
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1answer
89 views

Derivative of the Selberg $\zeta$-function

I want to compute the derivative of the Selberg $\zeta$-function: $$ \mathcal{Z}(s)=\prod_{\gamma \; \text{primitive}} \prod_{n=0}^\infty (1-e^{-l(\gamma)(n+s)}); \qquad \Re(s)>1.$$ Where ...
2
votes
1answer
304 views

Why is $\pi$ the Limit of the Absolute Value of the Prime $\zeta$ Function?

Motivation: I was looking at the approximation of the truncated Prime $\zeta$ function $$ P_x(s)=\sum_{p\leq x}p^{-s}= \mathrm{li}(x^{1-s}) + O \left(\cdot \right) $$ (to be found here with or ...
2
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1answer
63 views

Proof or source for this Hurwitz Zeta function identity?

I need a proof or source for this identity: $ \zeta '\left(z,\frac{q}{2}\right)-2^z \zeta '(z,q)+\zeta '\left(z,\frac{q+1}{2}\right)=\zeta(z,q)2^{z}\ln 2$ Here the derivative means the derivative by ...
2
votes
1answer
62 views

Evaluating Dirichlet series

It is well known that $$\eta(s)=\sum\limits_{k=1}^{\infty}\frac{(-1)^{k-1}}{k^s} =(1-2^{1-s})\zeta(s)$$ But I have the wider problem of evaluating the following ...
2
votes
2answers
443 views

Some basic questions about the Selberg zeta function

I'm trying to learn about the Selberg zeta function, but it seems like introductory texts assume more knowledge of Riemannian geometry than I'm comfortable with. I have some basic questions that ...
2
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1answer
45 views

How to solve the following delay differential equation?

What is the solution for the following equation? $$\frac\partial{\partial q}f(s,q)= \frac s2 f(s+2,q)$$ Note, it is known that the solution for $$\frac\partial{\partial q}f(s,q)= s f(s+1,q)$$ ...
2
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1answer
58 views

Residue of the (partial) Dedekind zeta-function

If $$\rho_\nu := res_{s=1} \zeta_K,\nu(s)=lim_{s\to 1}(s-1)\zeta_{K,\nu}(s)=\frac{2^r(2\pi)^s}{\omega_K |disc(\mathfrak(O)_K)|^{\frac{1}{2}}} \textrm{ (*)}$$ how can I follow that $$\rho := ...
2
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1answer
208 views

Convergence of the Fourier Transform of the Prime $\zeta$ Functions

I think I found a way to write the truncated Prime $\zeta$ function like this: $$ P_x(s)=\sum_{p<x} \frac{1}{p^s} =\sum_{n=1}^{\infty}\frac{ \mu (n)}{n} \sum_{z\in\{1,\rho\}}(-1)^{1-\delta_{1z}} ...
2
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1answer
261 views

Is there a Riemann hypothesis for the Hasse-Weil zeta function, generally? [duplicate]

What form does the Riemann hypothesis have for a global L-function?
2
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1answer
476 views

Is Riemann Zeta Function symmetrical about the real axis?

From wikipedia, http://en.wikipedia.org/wiki/Riemann_zeta_function "Furthermore, the fact that $\zeta(s) = \zeta(s^*)^*$ for all complex s ≠ 1 ($s^*$ indicating complex conjugation) implies that the ...
2
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1answer
23 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 ...
2
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0answers
44 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 \ | \ ...
2
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0answers
42 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 ...
2
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0answers
25 views

another representation of the zeta function of a curve over a finite field

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. ...
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0answers
60 views

Zeta zeros & Exponential Integral

From the oscillating part of an explicit formula for primes: $$\text{Re}(\operatorname{li}(x^\rho))\approx x\ \text{Re}(\operatorname{li}(x^{-\rho}))$$ $\rho_n$ may be replaced by any complex ...
2
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0answers
53 views

Intuitive explanation for $\zeta (2)=\frac{\pi^2}{6}$ [duplicate]

Using $f(x)=x^2$ Fourier' series, the proof for $\zeta (2)=\frac{\pi^2}{6}$ is pretty straight forward. I'm wondering if there is a more intuitive explanation for the equality, one that a layman could ...
2
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0answers
43 views

Relations betweens Multizeta Values

I am studying Multizeta values at the moment and I found that at weight 5, the basis is given by $\zeta(5)$ and $\zeta(3)\zeta(2)$ in the literature. Solving all shuffle and stuffle relations using ...
2
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0answers
24 views

Computing leading coefficient of $\zeta_K(s)$ at $s=0$

I am trying to prove the result stated in this Wikipedia page that $$\lim_{s\to 0} s^{-r} \zeta_K(s) = -\frac{h_k\cdot R}{w_K}$$ The formulae I have are: $$\begin{aligned}\xi_K(s) ...
2
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0answers
47 views

$\zeta(4)=\sum_ {k=1}^{\infty}{\frac{1}{k^4}}$ [duplicate]

How to Find $$\zeta(4)=\sum_ {k=1}^{\infty}{\frac{1}{k^4}}$$ the most basic way possible? I know it's $\pi^4/90$ but to arrive at this figure? Curious, because I need it to solve the integral ...
2
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0answers
79 views

Zeta Zeros and primes / prime powers

The plot $\Re\ x^{Zeta\ Zero} + \Im\ x^{Zeta\ Zero}$ for the first $1000$ Zeta Zeros up to $x = 30$ using the following Mathematica code: ...
2
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0answers
135 views

A Thue-Morse Zeta function ( Generalized Riemann Zeta function and new GRH )

Consider $t_n$ as the Thue-Morse sequence. Let $m$ be a positive integer and $s$ a complex number. Odiuos Number Now consider the sequence of functions below $f(1,s)=1+2^{-s}+3^{-s}+4^{-s}+...$ ...
2
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0answers
119 views

Determinant of the Laplacian of a surface is this correct?

given a surface with metric $ g_{ab} $ i would like to evaluate the functional determinant of the Laplacian in the form $ - \partial _{s} \zeta (0,E^{2})=\log\det( \Delta + E^{2}) $ then i need to ...
2
votes
0answers
92 views

Constant term of zeta binomials

Let's have the following zeta binomial $\sum\limits_{n=1}^\infty (1/n-1/(n+1))^k$, where $k$ a natural number and $k>1$. From the expansion of these binomials we obtain polynomials of $\pi$ where ...
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4answers
70 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 ...
1
vote
2answers
169 views

Series involving zeta functions

Let's have the following zeta function binomial: $\sum\limits_{n=1}^\infty \left(\frac{1}{n}-\frac{1}{n+1}\right)^2=\pi^2/3-3$. Does anyone know the limit of the following zeta function binomial ...
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vote
1answer
53 views

Pulsating waves of zeta function

Below is an animation of the partial sums of $\operatorname{li}x-2\Re\sum_{k=1}^{N}\operatorname{Ei}(\rho_k \log x)-\log2+\int_{x}^{\infty}\dfrac{\text{d}t}{t(t^2-1)\log t},$ for $1\leq N\leq100$ ...
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1answer
251 views

what the RH equivalent for Riemann prime formula $\Pi(x)$?

Question follow the one answered already, zeros about Riemann Zeta function and some L-function Let's me try my best to make it clear on what I am asking. In his 1859 paper "On the Number of Primes ...
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1answer
41 views

how to prove that $\zeta(s,q)=\frac{1}{\Gamma(s)}\int_{0}^{\infty}\frac{t^{s-1}e^{-qt}}{1-e^{-t}}dt$

how to prove that $$\zeta(s,q)=\frac{1}{\Gamma(s)}\int_{0}^{\infty}\frac{t^{s-1}e^{-qt}}{1-e^{-t}}dt:R(s)>1 , R(q)>0$$ where $\zeta(s,q)$ is Hurwitz zeta function
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1answer
114 views

Number of projective points on a curve

This is a continuation of this previous problem I asked about. From Ireland and Rosen's Number theory book(ch.11 ex#12) EDITED: Let $C_{1}$ be the curve $y^2=x^{3}-Dx$ over $\mathbb{F}_{p}$, where ...
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1answer
92 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 ...
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1answer
35 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 ...
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1answer
22 views

Notation for a zeta function

What does this notation mean: To provide some context, here are some of the exercises related to it: I initially thought the notation was such that $cos(2pi/n^n)+sin(2pi/n^n)$. This doesn't ...
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1answer
27 views

zeta function and probability divisible by k and choosing squares

I have some difficulty starting with this question: For $s>1$, set $\zeta(s)=\sum_{n=1}^\infty n^{-s}$, and define $p: \mathbb{N} \rightarrow [0,1]$ by $p(n)=\frac{n^{-s}}{\zeta(s)}$. One can ...
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2answers
144 views

On functions similar to Hurwitz zeta function

Denoted as $\zeta(s,a)$ for a > 0 Where do I find topics on the Hurwitz zeta function for a < 0? Any links or resources would be appreciated. (Please dont mention wiki or mathworld) Thanks
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1answer
45 views

Zeta Regularization and Products of Primes

How can one prove that: $2 * 3 * 5 * 7 \ldots = \prod_{n=1}^{\infty} p_i = 4\pi^2$ using zeta regularization? The sum diverges like the Ramanujan/Euler product but it can be associated to a value ...
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1answer
81 views

Difference between Meromorphic and Analytic Continuation

We have that $\frac{X}{spec \mathbb{Z}}$, a scheme of finite type. Consider $\zeta(X,s)= \prod_{x\in{X}} \frac{1}{(1-Nx^{-s}}$, with $Nx$ the norm. I didn't catch what my teacher was saying and he is ...
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1answer
121 views

proof regarding zeta function of a curve from Ireland and Rosen's “A Classical Introduction to Modern Number Theory”

In chapter $11$ section $5$ titled "the last entry" the authors state that the number of solutions to the congruence $x^2+y^2+x^2y^2 \equiv 1 \mod p$ is $p+1-2a$, where $p=a^2+b^2$ and $a+bi \equiv ...