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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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53 views

Prove $\sum_{n=1}^\infty \frac{\zeta(2n+1) - 1}{(2n+1) 2^{2n}} = 1 + \ln(2) - \ln(3) - \gamma$

I've found the following series on the Wikipediapage of the Euler-Mascheroni constant and I want to prove it. $$\sum_{n=1}^\infty \frac{\zeta(2n+1) - 1}{(2n+1) 2^{2n}} = 1 + \ln(2) - \ln(3) - \gamma$$...
1
vote
1answer
35 views

On the functions $\mathrm{Gi}_{s}^{p,q}(x)=\sum\limits_{n\geq0}\frac{x^{pn+q}}{(pn+q)^s}$

I have stumbled across the functions $$\mathrm{Gi}_s^{p,q}(x)=\sum_{n\geq0}\frac{x^{pn+q}}{(pn+q)^s}$$ And I would like to know where I can learn more about them. These functions are interesting ...
-1
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0answers
20 views

A taylor series for $\lim \limits_{h \to 0} \frac{\zeta'(-2n-h)}{\zeta(-2n-h)}$ [on hold]

Please help and use Big O Notation, thanks. I tried so-far making a Taylor Series for this limit but my solution is not what I seek for.
0
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1answer
18 views

Why is $res(\Gamma(s)x^{-s}\zeta(2s)|s=\frac{1}{2}) = \frac{\Gamma(\frac{1}{2})}{2x^{\frac{1}{2}}}$?

Why is $res(\Gamma(s)x^{-s}\zeta(2s)|s=\frac{1}{2}) = \frac{\Gamma(\frac{1}{2})}{2x^{\frac{1}{2}}}$? where res means the residue of the function? I know $\zeta(s)$ has a pole at $s=1$ but i can't ...
2
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0answers
29 views

Prove the series $\sum n^{-1-it}$ is diverge for all real $t$.

Prove that the series $\sum_{n=1}^\infty n^{-1-it}$ diverges for all real $t$. I have shown in the previous exercise that this series is bounded for nonzero $t$, and when $t=0$, it is famous that the ...
0
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1answer
21 views

Alternating Bertrand series

It is known that $$\frac{\partial^n\zeta(s)}{\partial s^n}=(-1)^n\sum_{k=1}^\infty{\frac{\log^nk}{k^s}}$$ Can the following alternating version of the sum be expressed in terms of well-known ...
0
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0answers
21 views

zeta distribution when s->1

Zeta distribution is just a set's density when s->1. I found this on the Wikipedia page about zeta distribution(https://en.wikipedia.org/wiki/Zeta_distribution). but I can't find a proof for it, can ...
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0answers
23 views

Expected value of zeta distribution [closed]

For what values of s, expected value of zeta distribution is finite? I know that the answer is s>2, but I'm not sure how to get there.
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1answer
28 views

Proof of Symmetry of zeros in critiical strip

How can we prove that zeros in critical strip are symmetric respect line $\sigma=\frac{1}{2}$ and respect real axis?
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1answer
59 views

Calculating Value of $\pi$ using Zeta Function on C++ spits error

I'm not sure if this is the correct place to ask this, but surely it's not a programming question. $$\zeta (s) = \prod \limits_{p \space prime} \left ( 1 - \frac{1}{p^s} \right)^{-1}$$ So, as we ...
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1answer
62 views

An analytic doubt from Riemann zeta function-Titchmarsh

Doubt form the Book Theory of Riemann Zeta Function by Titchmarsh I've problem on Theorem $5.8$ in the equation $5.8.2$. When $k=l$ the from Lemma $5.7$ we get, $$ \sum_{n=N+1}^bn^{-it}=O\left(N^{1-1/...
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2answers
135 views

Is it true that $\int_0^1 \big(K(k^{1/2})\big)^2\,dk = \frac{7}2\zeta(3)$?

Define the complete elliptic integral of the first kind as, $$K(k) = \tfrac{\pi}{2}\,_2F_1\left(\tfrac12,\tfrac12,1,\,k^2\right)$$ Part I. From the link above, we find some of the evaluations below, ...
1
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1answer
38 views

Error in counting the number of relatively coprime integer pairs less than N

The number of relatively coprime integers less than $N$ grows like $\frac{N^{2}}{\zeta(2)}$ (which, looking at the structure of $\frac{1}{\zeta(2)}=\Pi_{p_{k}}(1-\frac{1}{p_{k}^{2}})$, can be proven ...
2
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2answers
47 views

Where can I find papers and research on the Alternating Hurwitz Zeta Function?

The function is as follows (I don't know it's name, it could be 'Generalised Dirichlet Eta Function') $$f(s,q)=\sum_{n=0}^{\infty} \frac{(-1)^{n}}{(n+q)^{s}}$$
3
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1answer
82 views

Fixed point of Riemann Zeta function

I have been looking for fixed points of Riemann Zeta function and find something very interesting, it has two fixed points in $\mathbb{C}\setminus\{1\}$. The first fixed point is in the Right half ...
0
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1answer
20 views

Need help showing Riemann's Functional equation for negative numbers and complex numbers

Riemann's Functional equation: $\zeta(-z)$=${-2*z!\over(2\pi)^{z+1}}$$sin({\pi z\over2})$$\zeta(z+1)$This formulas expresses $\zeta(-z)$ in terms of $\zeta(z+1)$ Note: I read that the author said, ...
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1answer
21 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,...
0
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0answers
12 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
62 views

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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0answers
12 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 $...
3
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2answers
106 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 ...
3
votes
1answer
480 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
27 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
62 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
votes
1answer
87 views

On the logarithm of the fractional part Integral

Let $\{\}$ denote the fractional part function, then does the following integral admit a closed-form ? $$\int_{0}^{1}x\ln\bigg(\bigg\{\frac{1}{x}\bigg\}\bigg)dx$$
2
votes
1answer
95 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 ...
1
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1answer
113 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
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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 ...
2
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1answer
33 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
54 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}^{\...
4
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1answer
90 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
146 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
87 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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0answers
81 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
123 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
43 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
12 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
38 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}...
3
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1answer
50 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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0answers
27 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
115 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
40 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
0
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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}, ...
1
vote
0answers
91 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 ...
1
vote
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 > ...
5
votes
1answer
138 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[...