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Questions tagged [zeta-functions]

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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1answer
15 views

Computing the zeta function of curves over finite fields.

Given a nonsingular curve $X/\mathbb{F}_q$, how can one efficiently compute its zeta function $Z(X/\mathbb{F},T)$? My current strategy is to determine the genus $g$ of $X$, then count the points $N_1,...
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10 views

Average Error as Number of Samples Increases

I made a very simple program that approximates $\pi$ in r, by finding the probability that 2 random generated numbers are coprime for n trials. The result of this probability approaches $\frac{6}{\pi^...
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0answers
48 views
+50

using zeta function in the regularization of functional trace and determinant

I'm looking for books or lectures concerning Zeta function regularization. In particular, I'm interested in using zeta function in the regularization of functional trace and determinant . To be more ...
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10 views

Methods to extend the definition of eta to the entire complex plane

In this page (https://en.wikipedia.org/wiki/Dirichlet_eta_function) the author wrote: Hardy gave a simple proof of the functional equation for the eta function, which is: $$η(-s)=ϕ(s)η(1+s)$$ where $...
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2answers
80 views

Alternative proof for $\zeta\left(2,\frac14\right)=\psi^{(1)}\left(\frac14\right)=\pi^2+8G$

On the German Wikipedia page of the Hurwitz Zeta Function I have come across the following formula $$\zeta\left(2,\frac14\right)~=~\pi^2+8G\tag1$$ where $G$ is Catalan's Constant. Even though I ...
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0answers
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Zeta graph of values

Has anyone duplicated John Derbyshire's 'Prime Obsession' p.213 Zeta graph ? He only show values [on extreme right and left of graph] as they go to infinity but not X+Yi coordinates for those. The ...
3
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1answer
466 views

Are there known zeros of the Zeta function off the line 1/2?

I've been doing research in information theory that had some superficial connections to the Zeta function, since I was making use of logarithms of complex numbers. In short, we'd be looking at the ...
2
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0answers
26 views

$\chi$ varies over characters of $F$ of order dividing $m$, $\chi^{'}$ varies over characters of $F_s$ of order dividing $m$

A month ago I've asked two questions about rationality of the zeta function. The pages that belongs to my question are (linked here) Unfortunately I'm still clueless, but some steps are clear now. ...
2
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1answer
60 views

Solution to $\int_0^1 \left(\frac{\ln(x)}{1-x}\right)^2dx$ in a closed form. [duplicate]

I'm looking for the solution to the integral $$\int_0^1 \left(\frac{\ln(x)}{1-x}\right)^2dx$$ I solved and know that the solution to $$-\int_0^1 \frac{\ln(x)}{1-x}dx = \frac{\pi^2}{6}$$ through a ...
4
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1answer
83 views

On the logarithm of the fractional part Integral

Let $\{\}$ denote the fractional part function, then does the following integral admit a closed-form ? $$\int_{0}^{1}x\ln\bigg(\bigg\{\frac{1}{x}\bigg\}\bigg)dx$$
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1answer
80 views

Is there a connection between $\zeta(-1)$ and Ramanujan's calculation of the sum over $\mathbb{N}$?

Let me elaborate a little on the matter that I've been mulling over for a little while. This essentially concerns the summation of $1+2+3+...$, how it equals $-1/12$ (in a certain sense, obviously not ...
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1answer
104 views

An observation on Non-Trivial Zeros of Riemann Zeta Function.

I observed this property in month of July this year but unable to design a mathematical proof or mathematical way to state my observation. I need help to state this property. We know that for certain ...
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2answers
64 views

Find the residue of $\frac{\zeta'(1+s)}{\zeta(1+s)}\frac{x^s}{s}$ at $s=0$

Let, $\displaystyle f(s)=\frac{\zeta'(1+s)}{\zeta(1+s)}\frac{x^s}{s}$. Prove that $Res(f,s=0)=A-\log x$ , for some constant $A$. At $s=0$ , $\zeta(s)$ has a pole of order $1$ and $\zeta'(s)$ has a ...
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1answer
31 views

Zeta like function summing over Gaussian integers in the first quadrant

Let $x$ be a real number and let $$f(x)=\sum_{ z = re^{\theta i} \in\mathbb{Z}[i] \\ r \le x \\ 0\le \theta \le \pi/2}\frac{1}{z^s}$$ Is it possible to compute in (terms of the $\zeta$ function ...
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2answers
53 views

Is $2 \sum_{n=m+2}^{\infty}\dfrac{1}{n^{\alpha}} \geq \sum_{n=m+1}^{\infty}\dfrac{1}{n^{\alpha}}$?

Working in a project I came across to the following problem, to investigate if the following inequality is true for $\alpha>2,$ and $m$ a positive integer: $$2 \sum_{n=m+2}^{\...
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1answer
60 views

Is this zeta-type function meromorphic?

In An older question I asked : ( See 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 $s$ ...
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1answer
144 views

Calculating $S=\sum\limits_{n=1}^\infty\left(\frac{1}{\Gamma^2(n+1)}\right)^{{1}/{n}}$

I tried to find the answer for the question: Numerical evaluation of $\sum_{N=1}^\infty\left(\frac{1}{\Gamma(N+1)^2}\right)^{\frac{1}{N}}$. I think my result is $4$ times than the expected value. Is ...
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0answers
74 views

A computation in Conrey's paper on Riemann zeta function

I am reading the Conrey's paper "More than two fifths of the zeros of the Riemann zeta function are on the critical line" (see here). I have question/doubt in a particular step: In P.10, it claimed ...
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78 views

Proving that $\zeta(2)$ is irrational.

We have the following integral: \begin{align} I_n=\int_0^1 P_n(x)\frac{\ln x}{1-x}dx = \frac{a_n\zeta(2)+b_n}{d_n^2} \end{align} $P_n(x)$ is a polynomial with integer coefficients and degree $n$. $a_n,...
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0answers
122 views

Trying to understand why the zeta function is a rational function under certain conditions. Questions about some equations.

Information: I linked the pages below, which relate to my questions. I am currently reading " A Classic Introduction to Modern Number Theory " by Kenneth Ireland and Michael Rosen. In the 11th ...
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1answer
90 views

Question about characters ( Section : The Rationality of the Zeta Function associated to $a_0x_0^m+a_1x_1^m+…+a_nx_n^m$ )

I am currently reading " A Classic Introduction to Modern Number Theory " by Kenneth Ireland and Michael Rosen. In the 11th chapter they consider the zeta function. In the third section of this ...
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0answers
42 views

An inequality involving the Dirichlet eta function

I would be interested in proving the following inequality involving Dirichlet's eta function $\eta(s)$ at different values which, after some numerical investigations, I am sure is true $|\chi(1-s)\...
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0answers
16 views

Error in expression of incomplete zeta function?

Background & Question I realised I could do a different manipulation from the one I did over here Strange method to obtain strange number theoretic identities? However after making some ...
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0answers
11 views

Limit of local minimum of function involving zeta function ($e\log x$)

I came across the following interesting limit $$\lim\limits_{n\to\infty}\left(\min(x^{n+1}\zeta(x))-\min(x^{n}\zeta(x))\right)$$ for $x\in\mathbb{R}$ and $x\gt1.$ Calculation highly suggests the ...
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1answer
33 views

Order of $\Gamma(n/2)$

I want to estimate the order of $\Gamma(n/2)$. We have from Stirling interpolation , for sufficiently large value of $n$, \begin{align} \Gamma(n/2)&\approx \sqrt{\frac{4\pi}{n}}\left(\frac{n}{2e}...
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1answer
45 views

Proof of $ \zeta(s)=\frac{1}{s-1}+\gamma+O(s-1)$

Prove that $\displaystyle \zeta(s)=\frac{1}{s-1}+\gamma+O(s-1)$,near $s=1$, where $\gamma$ is Euler's Constant. I've proved $\displaystyle \zeta(s)=s\int_1^{\infty}\frac{[x]-x+1/2}{x^{s+1}}\,dx+\frac{...
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25 views

Superscript numbers on a Hurwitz Zeta function (Wolfram Alpha)

http://www.wolframalpha.com/input/?i=x(n%2B1)%3Dsqrt(n%2F(n%2B2))*x(n) See the x(n) solution given - there is a Zeta function with a (0,1) superscript, which I'm unfamiliar with. Additionally, the ...
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1answer
48 views

How to prove that $3^{x^2+x} (x+1)^{-x} \Gamma (x+1)\ge 1$ for $x>0$?

Let $$f(x)=3^{x^2+x} (x+1)^{-x} \Gamma (x+1).$$ Drawing a picture with any computer algebra system, it is obviously that $f(x) \ge 1$ on $[0,\infty)$. But How can we prove this? If we take ...
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1answer
100 views

How to show that $\int_0^1 \sin \pi t ~ \left( \zeta (\frac12, \frac{t}{2})-\zeta (\frac12, \frac{t+1}{2}) \right) dt=1$?

I've been trying to prove Fresnel integrals by real methods and encountered an interesting problem. Let's start with the known result: $$\int_0^\infty \sin y^2 dy = \sqrt{\frac{\pi}{8}}$$ Can we ...
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1answer
38 views

Alternating series of 'zeta' functions for real integer parameters equals 1/2

I would like to know how to show that: $$ \sum_{s=2}^{\infty}(-1)^s\zeta(s)=\dfrac{1}{2} $$ where $$ \zeta(s)=\sum_{n=1}^{\infty}\dfrac{1}{n^s} \\s,n \in N $$ Kind regards
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1answer
42 views

How Riemann made Z(s) to converge for s>0?

I am going through the procedure that Riemann took to expand the Euler's Z-funtion. But I can not understand first part of it. We know that Z(s) just converge for s>1. How it is possible to make ...
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0answers
42 views

A question about the rationality of Hurwitz Zeta functions.

I'm checking over some of my work from studying prime numbers, and I found where I used the Hurwitz Zeta function $$\zeta(s,\alpha)_{\alpha \in [0,1)} = \sum_{n\in \mathbb{N}} \frac{1}{(n+\alpha)^s}, ...
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0answers
79 views

Deriving the closed form of Gamma function using Euler-Chi function

Background #1 Here is a part of an answer of @Sankyu Kim in MathOverflow. Consequently, we get the Euler-chi function $\chi(z):=\frac{\zeta(1-z)}{\zeta(z)}$. And I want to know if Sankyu Kim's ...
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0answers
29 views

Epsteins zeta function

The Epstein's zeta function is: $Z(Y,s)=\underset{0\neq a\in\mathbb{Z}^{n}}{\sum}(a^{t}Ya)^{-s}$, where Y is a positive symmetric difinite $n \times n$ matrix. Why does it converges when $\sigma > ...
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1answer
117 views

The prime number theorem over a finite field - Lang's *Algebra*, Chapter V, Exercise 23(b)

This is Exercise 23(b) of Chapter V (Algebraic Extensions) from Lang's Algebra. Let $k$ be finite field with $q$ elements, and let $\pi_q(n)$ be the number of monic irreducible polynomials $p \in k[...
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24 views

Zeta function in graph theory

I would like to know about the applications of Zeta function in graph theory. Thanks!
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57 views

Convergence of double sum

I'm thinking this has to do with dominated convergence. I know it kinda works, but I'm not 100% sure why and how the steps are rigorously carried out and possibly the above theorem is applied. Let me ...
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1answer
88 views

When is $(\frac{v_1}{1}+\frac{v_2}{2}+ \dots \frac{v_p}{p})+(\frac{v_1}{p+1}+\frac{v_2}{p+2}+\dots \frac{v_p}{2p})+\dots$ equal to zero?

Let $V=\{\vec{v} =(v_1,\dots v_p): v_i\in \mathbb{Z}\}$. And define $$\phi(\vec{v},s)= \sum_{n=1}^\infty\frac{v_n}{n^s}$$ Where for all $k=1,2, \dots$ we have $v_{k+p}=v_k$. My question (1) Is ...
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79 views

Prime twins and $ f(z) = \sum_p \frac{\ln(p) \space \ln(p+2)}{p^{z/2} \space (p+2)^{z/2}} $

Let $p$ be a prime such that $p+2$ is Also a prime. Define $$ f(z) = \sum_p \frac{\ln(p) \space \ln(p+2)}{p^{z/2} \space (p+2)^{z/2}} $$ For $z \neq 1 $ and $re(z) > 1$ this $f(z)$ always ...
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63 views

Alternating series combining harmonic number and zeta values

While evaluating the following fractional part integral, I get stuck on an almost euler sum as highlighted in red colour. Could someone evaluate the red series in terms of well-known constants ? $$\...
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5answers
264 views

Double series convergent to $2\zeta(4)$?

Using a computer I found the double sum $$S(n)= \sum_{j=1}^n\sum_{k=1}^n \frac{j^2 + jk + k^2}{j^2(j+k)^2k^2}$$ has values $$S(10) \quad\quad= 1.881427206538142 \\ S(1000) \quad= 2.161366028875634 \\...
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0answers
20 views

Relation between infinite product and regularized product

For a positive sequence $0\le\lambda_{1}\le\lambda_{2}\le\cdots$, consider an infinite product \begin{equation*} \prod_{i=1}^{\infty}\lambda_{i}:=\lim_{n\rightarrow\infty}\prod_{i=1}^{n}\lambda_{i}...
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3answers
311 views

On the sets of sums $\sum\limits_{n=1}^\infty\frac{a_n}{n^s}$ with $(a_n)$ periodic and integer valued, for different values of $s$ natural number

For every positive integer $s$, let $A_s$ denote the set of the sums of the converging series $\sum\limits_{n=1}^\infty\frac{a_n}{n^s}$ for every periodic sequence of integers $(a_n)$. Then each $A_s$...
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0answers
31 views

On the Puiseux series of divergent zeta function for $0 < \Re(s)< 1$

Let $s$ be a complex number such that 0 < $\Re(s) < 1$ and $\zeta(s)$ be the analytic continuation of zeta function on the strip 0 < $\Re(s) < 1$. Then by applying the Euler-Maclaurin ...
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0answers
62 views

Convergence of alternating zeta function [duplicate]

I have been studying the Riemann zeta function and I came across such a representation of zeta function $\zeta(s)=(1-2^{1-s})^{-1} \sum_{n=1}^{\infty} (-1)^{n-1} n^{-s}$. Then it says in the book ...
2
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1answer
58 views

On the sum of $\sum_{p \ prime} \frac{1}{p^2-1}$ [closed]

I was wondering whether there exists a closed form solution for $\sum_{p \ prime} \frac{1}{p^2-1}$?
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3answers
133 views

What's the series of $\sum_{n\geqslant1} \dfrac{\zeta(2n)}{n2^{2n}}$.

I know with the formula $$1-\sum_{n\geq 1}2\zeta(2n)\,x^{2n}=\pi x\cot(\pi x)$$ may I find the following relation used here $$ \sum_{n\geqslant1} \dfrac{\zeta(2n)}{n2^{2n}}=\color{blue}{\ln\dfrac{\...
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1answer
62 views

Evaluation of a series involving the harmonic number

Let $\text{H}_n$ denote the n-th harmonic number, does the following sequence have a closed-form ? As an approximation I have got $0.0922514...$ $$ \frac{3}{2}+\lim_{n\to\infty} \bigg(\sum_{k=3}^{n}\...
2
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3answers
122 views

Square root of fractional part integral

Does the following integral have a closed form ? $$\int_{0}^{1}\sqrt{\bigg\{\frac{1}{x}\bigg\}}dx$$ Where $\{x\}$ denotes the fractional part of $x$.
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0answers
47 views

Set of poles of Igusa zeta functions at large primes

Let $\Omega_p$ denote the set of the real parts of the poles of the Igusa zeta function of a polynomial $f\in$Z$[X_1,\dots,X_m]$ (you may assume $f$ homogeneous if it helps) at the prime $p$. I am ...