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Questions tagged [zeta-functions]

Questions on various generalizations of the Riemann zeta function (e.g. Dedekind zeta, Hasse–Weil zeta, L-functions, multiple zeta). Consider using the tag (riemann-zeta) instead if your question is specifically about Riemann's function.

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What is the identity of this zeta function?

There are a Riemann zeta function, a Hurwitz zeta function, and many different types of zeta functions. However, I saw the zeta function below in a Japanese blog. $$\zeta(s)=\frac{1}{1-2^{1-s}}\sum_{m=...
user1274233's user avatar
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is this solution correct $\frac {\partial}{\partial x} \int_0^∞ \frac{\sin((x+it)\arctan(t))}{((1+t^2)^{(x+it)/2} (e^{2\pi t} -1))} dt =0 $?

when I was reading about the Riemann zeta function I found out this integral $\ \frac {\partial}{\partial x} \int_0^∞ ​ \frac{\sin((x+iy)\arctan(t))}{((1+t^2)^{(x+iy)/2} (e^{2\pi t} -1))} dt $ and ...
Prateek Sharma's user avatar
-3 votes
0 answers
19 views

what is the value of d/ds (integral(0,infinity) sin(s arctan t)/((1-t^2)(s/2) (e^(2pi t) -1)) dt) [closed]

I was reading about the Riemann zeta function and found the integral form of it ζ(s)= 1/(s-1) + 1/2 + 2(integral(0, infinity) sin(s arctan t)/((1-t^2)(s/2) (e^(2pi t) -1)) dt). When I was trying to ...
Prateek Sharma's user avatar
1 vote
0 answers
57 views

Periodic zeta function

Let $e(x)=e^{2\pi ix}$ and let $$F(x,s)=\sum _{n=1}^\infty \frac {e(nx)}{n^s}$$ be the periodic zeta function. What is the functional equation for the periodic zeta function ?: I can find a statement ...
tomos's user avatar
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1 vote
1 answer
109 views

Finding a closed form for $\sum^{\infty}_{n=1} \frac{1}{(n+1)n^\alpha}$ [duplicate]

I encountered the following sum in my work and I was wondering if it has a known closed form: $$ \sum^{\infty}_{n=1} \frac{1}{(n+1)n^{\alpha}} \quad , \quad 0 < \alpha < 1 \; , \; \alpha \in \...
Aidan R.S.'s user avatar
8 votes
3 answers
963 views

"Are there any simple groups that appear as zeros of the zeta function?" by Peter Freyd; why is this consternating to mathematicians?

I would like to understand the "upsetting"-to-mathematicians nature of this question Freyd poses to demonstrate that "any language sufficiently rich that to be defined necessarily ...
Hooman J's user avatar
  • 247
3 votes
1 answer
94 views

Closed form for $A = \sum_{a>1,b>1}\dfrac{1}{(2 a^2 + 3 b^2)^2}$?

Is there a closed form for $A = \sum_{a>1,b>1}\dfrac{1}{(2 a^2 + 3 b^2)^2}$ ?? We know $$ \sum_{m,n = - \infty}^{\infty} \frac{(-1)^m}{m^2 + 58 n^2} = - \frac{\pi \ln( 27 + 5 \sqrt {29})}{\sqrt {...
mick's user avatar
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9 votes
3 answers
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How does Wolfram Alpha know this closed form?

I was messing around in Wolfram Alpha when I stumbled on this closed form expression for the Hurwitz Zeta function: $$ \zeta(3, 11/4) = 1/2 (56 \zeta(3) - 47360/9261 - 2 \pi^3). $$ How does WA know ...
Klangen's user avatar
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0 answers
19 views

Extensions of Hardy's Inequality for Tail Sums

I have been studying various formulations of Hardy's inequality, inspired by notable theorems from works like those of Paul Richard Beesack and others. A particular theorem that caught my attention ...
Snowball's user avatar
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0 answers
28 views

What's the point of the local zeta function?

I'm currently reading through Ireland and Rosen's "A Classical Introduction to Modern Number Theory", and I feel I'm missing the point of Chapter 11 on (local) zeta functions. When I see ...
Samuel Johnston's user avatar
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1 answer
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Estimation of the absolute value of the $n$th non-real zero of the Riemann zeta function

Recently, I have been studying the oringinal proof of the prime number theory by Hadamard. I didn't get it on the estimation of the absolute value of the $n$th non-real zero of the $\zeta$ function by ...
Derek Xie's user avatar
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0 answers
72 views

the zero's of $f(s,a) = \sum_{n=1}^{a-1} n^{-s} $

I was looking at the zero's of $$f(s,a) = \sum_{n=1}^{a-1} n^{-s} $$ for integer $a>3$ in the strip $0 < \operatorname{Re}(s) < 1$. Now this clearly relates to the Riemann zeta: $$f(s,a) + \...
mick's user avatar
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-1 votes
1 answer
55 views

Some curios sums of Hurwitz zeta-function and Lehmer's totient problem [closed]

For all squarefree $k\in \mathbb N$ $$\left|\frac{\sum_{n=1}^{k-1}\zeta_H(-1,n/k)}{\sum_{n=1}^{k-1}\chi_0(n)\zeta_H(-1,n/k)}\right|=\left|\frac{\sum_{n=1}^{k-1}1}{\sum_{n=1}^{k-1}\chi_0(n)}\right|=\...
user714's user avatar
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0 answers
42 views

Question about a limit of the euler-product of the riemann zeta function

The limit $$\lim_{s\to 1} (s-1)\zeta(s)=1=\lim_{s\to 1}(s-1)\prod_{p\in\mathbb P} (1-p^{-s})^{-1}$$ is well-known. Consider that there are infinitely many distinct subsets $\mathbb P_{k}\subsetneq\...
user714's user avatar
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4 votes
1 answer
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Showing that $\int_0^1 \frac{\text{d}x}{\zeta(x)\Gamma(x)}<0<1<\int_0^\infty \frac{\text{d}x}{\zeta(x)\Gamma(x)}$ in $3$ minutes, without a calculator

The following question is to be solved within $3$ minutes, without a calculator. $$\text{Let }I=\int_0^\infty \frac{\text{d}x}{\zeta(x)\Gamma(x)}\text{, and let }J=\int_0^1 \frac{\text{d}x}{\zeta(x)\...
Hussain-Alqatari's user avatar
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0 answers
35 views

Estimates of the derivatives of $\Xi(s)$

The $\Xi$ Function is defined by $\Xi(s)=\xi(\frac{1}{2}+is)$, where $\xi(s)=\frac{1}{2}s(s-1)\pi^{-\frac{s}{2}}\Gamma(\frac{s}{2})\zeta(s)$. This is a problem from my homework: since we can write it ...
Fresh's user avatar
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0 answers
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Zeta Hurwitz function bounds of summation

I have been trying to derive the following equality $$ \frac{1}{2} \sum_{n=1}^{\infty} \sum_{m=0}^{p-1} \frac{1}{n^s}\left[\cos \left(2 \pi \frac{m q}{p}\left(q^*-1+n\right)\right)+\cos \left(2 \pi \...
faker 23's user avatar
0 votes
1 answer
55 views

How can I evaluate $1 / \Gamma (-1)$?

I understand intuitively that the Gamma function diverges at all negative integers. This leads me to believe that the inverse Gamma function would have zeros at all negative integers. However, I’m ...
philipp nirnberger's user avatar
0 votes
1 answer
95 views

Showing $\sum_{n=1}^{\infty }\left ( \sum_{j=1}^{\infty }\frac{x^{(n-j)^2}-x^{(n+j-1)^2}}{(2n-1)(2j-1)} \right ) = \frac{\pi^2}{8}$

Show that $$\sum_{n=1}^{\infty }\left ( \sum_{j=1}^{\infty }\frac{x^{(n-j)^2}-x^{(n+j-1)^2}}{(2n-1)(2j-1)} \right) = \frac{\pi^2}{8}$$ I liked this problem because the result is a final answer, and ...
Dmitry's user avatar
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3 votes
2 answers
230 views

Is there a closed form expression for $ \sum_{n=2}^\infty \frac{1}{n \sqrt{n^2-1}} $?

I tried to find a closed form for the series $$ \sum_{n=2}^\infty \frac{1}{n \sqrt{n^2-1}} $$ I got another form for the series by using the known series $$\frac{1}{\sqrt{n^2-1}}=\frac{1}{n\sqrt{1-n^{-...
Faoler's user avatar
  • 1,657
5 votes
1 answer
271 views

Relationship between $\zeta(3)$ and ordinary logarithm function $\text{log}(\text{x})$

While working on another problem, I came up with the following expression. This involved many manual definite integral evaluations and not at all elegant. So I am not going into the details of the ...
Srini's user avatar
  • 862
0 votes
0 answers
62 views

Evaluating the sum $\sum_{n=1}^\infty \frac{1}{n(n+a)^b}$ [duplicate]

I am looking for ways to simplify the sum $$\sum_{n=1}^\infty \frac{1}{n(n+a)^b}, \quad a\in\mathbb{R}^+, b\in\mathbb{N}.$$ The first thought I had approaching this was to use Hurwitz and/or Zeta ...
minimax's user avatar
  • 45
0 votes
0 answers
74 views

Difficulty computing $\int_{0}^{\infty} \frac{\ln(x)}{e^x+1} dx$=$-\frac{1}{2}\ln^2(2)$ [duplicate]

Here some context , after computing some integral in the form $\int_{0}^{\infty} \frac{\ln(a^2+x^2)}{\cosh(x)+\cos(b)}dx \,\,\, , \int_{0}^{\infty} \frac{x\ln(a^2+x^2)}{\sinh(x)}dx \,\,\, , \int_{0}^{\...
azur's user avatar
  • 167
2 votes
1 answer
104 views

Limit of a Function Involving Hurwitz Zeta Function

I am trying to prove the following limit of a function involving the Hurwitz Zeta function: $$ \lim_{N \to \infty} \frac{\zeta(-d, 1 + N) - \zeta(-d, 1 + p N)}{N^{1 + d}} = \frac{-1 + p^{1 + d}}{1 + ...
Amirhossein Rezaei's user avatar
0 votes
2 answers
189 views

Proper Way to Calculate Value of Riemann Zeta function?

I understand that an Analytic Continuation of a function will extend its domain into areas that it previously wasn't defined in. I've been looking at one of the Analytic Continuations of the Zeta ...
Mods And Staff Are Not Fair's user avatar
2 votes
0 answers
127 views

Is there other method to deal with $\int_0^1 \int_0^1\frac{\ln^m (1+x y)}{(1+x y)^n}d x d y,$ where $m\ge 3?$

After finding the exact value of the integral, $$\displaystyle \int_0^1 \int_0^1\left(\frac{\ln (1+x y)}{1+x y}\right)^2 d x d y=-\frac{\zeta(3)}{4}+\frac{1}{3} \ln ^3 2-\frac{\pi^2}{6}+\ln ^2 2+2 \...
Lai's user avatar
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0 votes
0 answers
30 views

Zeta functions on q-deformed compact Lie groups

I’m reading some of the recent works on representation zeta functions for groups. Along the way, I have also explored some of the remarkable properties of Witten zeta function. I’m wondering if there ...
TMRiddle's user avatar
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0 answers
39 views

Relations between Dilogarithms and Imaginary part of Hurwitz-Zeta function

I'm working through a paper that involves a problem concerning the calculation of the Imaginary part of the derivative of the Hurwitz-Zeta function $\zeta_H(z,a)$ with respect to $z$, evaluated at a ...
MultipleSearchingUnity's user avatar
1 vote
1 answer
59 views

Connection between the polylogarithm and the Bernoulli polynomials.

I have been studying the polylogarithm function and came across its relation with Bernoulli polynomials, as Wikipedia site asserts: For positive integer polylogarithm orders $s$, the Hurwitz zeta ...
Dr Potato's user avatar
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0 votes
1 answer
56 views

step function question: What tools can be used to study it?

Consider the step function $$A(x)=\sum_{n=1}^\infty e^{\mathrm{floor}\bigg(\frac{\log n}{\log x}\bigg)+\mathrm{floor}\bigg(\frac{\log n}{\log (1-x)}\bigg)} = \prod_{\mathrm{ p~ prime}} \frac{1}{1-e^{\...
zeta space's user avatar
0 votes
1 answer
165 views

How to subtract two infinite series?

While applying the Sieve of Eratosthenes to Zeta function we subtract $\frac{1}{2^s} \zeta(s)$ from $\zeta(s)$ to obtain $$\zeta(s) - \frac{1}{2^s}\zeta(s) = \sum \frac{1}{n^s} - \sum \frac{1}{(2n)^s}$...
zeynel's user avatar
  • 447
1 vote
3 answers
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What are some unique integral representations of Apery's constant - $\zeta(3)$?

I've been playing around with some integrals, and I started looking at Apery's constant. There are some integral representation I've found online, such as: $$\zeta(3)=\frac{16}{3}\int_{0}^{1}\frac{x\...
Abdullah's user avatar
4 votes
3 answers
136 views

I need help evaluating the integral $\int_{-\infty}^{\infty} \frac{\log(1+e^{-z})}{1+e^{-z}}dz$

I was playing around with the integral: $$\int_{-\infty}^{\infty} \frac{\log(1+e^{-z})}{1+e^{-z}}dz$$ I couldn't find a way of solving it, but I used WolframAlpha to find that the integral evaluated ...
Abdullah's user avatar
0 votes
0 answers
59 views

What are these points of lower magnitude in the Riemann Zeta?

I recently visualized the Riemann Zeta function and noticed an interesting pattern: there are points of slightly lower magnitude along a very slightly curved vertical line, extending furthest out to ...
Some Guy's user avatar
1 vote
0 answers
50 views

A problem with Mertens Theorem

According to Mertens Theorem $$ \lim_{n \to \infty } (\frac{1}{ln(p_n)} \prod_{k=1}^n \frac{1}{1-\frac{1}{p_k}})=1.781072\dots$$ so we can say $$ 1.78 \ ln(p_n) \sim \prod_{k=1}^n \frac{1}{1-\frac{1}{...
kodar 's user avatar
  • 63
1 vote
1 answer
101 views

An infinite series of powers of fractions.

The series at hand is given by, $$\sum_{k=1}^{\infty} \left(\frac{2k+1}{k (k+1)}\right)^s$$ I think it converges for $s>1$, but I have not been able to derive a general expression for this series. ...
mathphy24's user avatar
1 vote
0 answers
20 views

Where are the non-real zero's of this Zeta like function $\prod_p \dfrac{1}{1 - p^{-s}} + \prod_q \dfrac{1}{1 - q^{-s}} $?

Let $s$ be a complex number. Define $\zeta(s)$ as the analytic continuation of $$\zeta(s) = \frac{\prod_p \dfrac{1}{1 - p^{-s}} + \prod_q \dfrac{1}{1 - q^{-s}}}{2} $$ where $p$ are the primes $1 \mod ...
mick's user avatar
  • 16.4k
1 vote
0 answers
46 views

Is there a useful/meaningful notion of a multi-variable L-function in number theory?

I recently encountered multi-variable generalizations of various classical zeta functions. For example, the multi-variable Riemann zeta function $$ \zeta(s_1, \ldots , s_r) := \sum_{0 < n_1 < \...
xion3582's user avatar
  • 470
0 votes
0 answers
33 views

Show $\lim_{x\rightarrow+\infty}\zeta(-a,x-a)+\zeta(-a,x+a)-2~\zeta(-a,x)=0.$

$$\lim_{x\rightarrow+\infty}\zeta(-a,x-a)+\zeta(-a,x+a)-2~\zeta(-a,x)=0$$ I was trying to solve an infinite summation problem of quadratic radicals, in which I used the Hurwitz Zeta Function to try to ...
spacedog's user avatar
  • 373
2 votes
1 answer
201 views

Evaluate $\int_{0}^{\frac{\pi}{2}} \theta \log^2(\cos\theta) d\theta$

$$\int_{0}^{\frac{\pi}{2}} \theta \log^2(\cos\theta) d\theta$$ $$\log(\sin(\theta)) \cdot \log(\cos(\theta)) = \frac{1}{4} \zeta(3) - \log(2) - \sum_{n=1}^{\infty} f(n) \cos(2n\theta)$$ and then we ...
user avatar
2 votes
1 answer
103 views

$|\zeta(1/2 + it)|^2 \geq \frac{\log(t)}{\log \log(t)}$

I've tried to solve this exercise for hours but I didn't managed to figure it out. Show that there exists a sequence $t \to \infty$ for which $$|\zeta (1/2 + it)|^2 \geq \frac{\log(t)}{\log\log(t)}$$ ...
Paul's user avatar
  • 1,374
-1 votes
1 answer
81 views

$\sum _{k=1}^{\infty }{\frac {\coth(k\pi )}{(k\pi )^{4n-1}}}$ [duplicate]

Show that $${\displaystyle \sum _{k=1}^{\infty }{\frac {\coth(k\pi )}{(k\pi )^{4n-1}}}=\sum _{k=0}^{2n}(-1)^{k-1}\,{\frac {\zeta (2k)}{\pi ^{2k}}}\,{\frac {\zeta (4n-2k)}{\pi ^{4n-2k}}}\qquad n\in \...
user avatar
1 vote
1 answer
78 views

$\frac 1{2\pi i}\int_{2-\infty i}^{2+\infty i}\frac{x^s}{s^2}\left(-\frac{\zeta^{\prime}(s)}{\zeta(s)}\right)ds$ as a finite sum of $\Lambda(n)$

[Introduction to Analytic Number Theory - Tom M. Apostol, chapter 13, question 7] Express $$\frac 1{2\pi i}\int_{2-\infty i}^{2+\infty i} \frac{x^s}{s^2}\left(-\frac{\zeta^{\prime}(s)}{\zeta(s)}\right)...
Sayan Dutta's user avatar
  • 9,592
1 vote
0 answers
78 views

For arithmetical periodic function $f$, if $\sum_{r=1}^k f(r)=0$, then $S=\sum_{n=1}^\infty \frac{f(n)}{n^{s}}$ converges

[Introduction to Analytic Number Theory - Tom M. Apostol, chapter 12, question 1(b)] Let $f(n)$ be an arithmetical function which is periodic mod $k$. If $$\sum_{r=1}^k f(r)=0$$ then prove that the ...
Sayan Dutta's user avatar
  • 9,592
1 vote
0 answers
25 views

What does $(Z - \delta)^k [x,y]$ count?

I am trying to understand my professor's lecture notes about zeta function and using it to count. DEFINITIONS The zeta function is defined as $Z[x,y] = \begin{cases} 1 & \text{if > } x \leq ...
Zek's user avatar
  • 309
0 votes
0 answers
62 views

"Mollifier" of the Dirichlet L-function

I was studying some zero-density results for $\zeta(s)$, mostly from Titchmarsh's book "The Theory of the Riemann zeta function", Chapter 9. In one place, as per the literature, a mollifier ...
djangounchained0716's user avatar
2 votes
1 answer
89 views

Proofs involving manipulation of divergent series

Is this proof valid even though the harmonic series it is based on is divergent? Prove: $$\sum_{n=2}^\infty (\zeta(n)-1)= 1$$ Where $\zeta$ as in Riemann's Zeta function is summed over all natural ...
Older Amateur's user avatar
0 votes
0 answers
20 views

$ f(s) = 1 + \sum_{n=2}^{\infty} \frac{n^{-s}}{p_n}= 0$?

Let $p_n$ be the $n$ th prime number. Let $f(s)$ be a Dirichlet series defined on the complex plane as : $$f(s) = 1 + \sum_{n=2}^{\infty} \frac{n^{-s}}{p_n}= 1 + \frac{2^{-s}}{2}+ \frac{3^{-s}}{3} + \...
mick's user avatar
  • 16.4k
-1 votes
1 answer
279 views

Explore the relationship between $\sum\limits_{n = 1}^{2x} \frac{1}{{n^s}^x}$ and $\sum\limits_{n = 1}^{2x-1} \frac{(-1)^{n-1}}{{n^s}^x}$ [closed]

I am trying to find an algorithm with time complexity $O(1)$ for a boring problem code-named P-2000 problem. The answer to this boring question is a boring large number of $601$ digits. The DP ...
user avatar
0 votes
0 answers
32 views

A certain formal power series equals the zeta function of a curve

Be $X$ a projective, geometrically connected and smooth curve over the finite field $\mathbb{F}_q$. Then one can define the following series, i.e. the zeta function of the curve: $$Z(t) = \exp\left(\...
Luca Morstabilini's user avatar

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