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.

Filter by
Sorted by
Tagged with
3 votes
1 answer
53 views

General formula for logarithmic cosine integral

I am trying to find a general expression for $$ \int_0^{\pi/2} (\ln(\cos(x)))^n dx $$ for integer $n$, but have not been able to find it online or derive it. I have a good idea that the general form ...
user avatar
2 votes
0 answers
21 views

Asymptotically similar functions with opposite parity, were they considered, are they useful? Case of polynomials

So, can we transform an even function into an odd function and vice versa? Let's consider this method: Transformation even->odd: Suppose $f_{even}(x)$ is a function which satisfies the following ...
user avatar
  • 7,452
0 votes
0 answers
17 views

Estimation of the Totient Summatory Function

The following statement appears on the Wikipedia page for the Totient Summatory Function: The Totient summatory function is defined by: $\Phi(n):=\sum_{k=1}^n\varphi(k),\ \ \ \ n\in\mathbb{N}$ Using ...
user avatar
  • 1
0 votes
0 answers
35 views

A question on the relation between the complex zeroes of zeta function and the estimate of the error in PNT

I'm currently working on the following problem from Analytic Number Theory. Assume that $\psi(x)-x=\mathcal{O}(x^a)$, for some $1/2<a<1$, where $\psi$ is the Chebyshev function. I would like to ...
user avatar
  • 27
8 votes
1 answer
142 views

Best bound for $|\sum \prod_{p\mid n}(1-\frac{1}{p^k})-\frac m{\zeta(k+1)}|$?

Follow from the previous post, we have \begin{align*} f_2(x):=\sum_{n\le x} \prod_{p\mid n}(1-\frac{1}{p^2})-\frac x{\zeta(3)}=O(1)\quad \text{as }x\to\infty.\tag{*} \end{align*} When I tried to plot $...
user avatar
  • 5,027
0 votes
0 answers
36 views

inequality for the zeta function for real $s$

I am wondering why $\zeta(s)<1+\frac{1}{2^s}+(s-1)^{-1} 2^{-s+1}$ for $s>1.$ I was able to show that $\zeta(s) \leq \frac{1}{1-\frac{1}{2^{s-1}}}$ but thats a much bigger upper bound. Does ...
user avatar
  • 9
2 votes
1 answer
85 views

Is there any other method to compute $\int_{0}^{\frac{\pi}{4}} y \ln (\cos y) d y$?

After investigating the integral $$ \int_{0}^{\frac{\pi}{2}} y \ (\cos y) d y $$ in the post. I keep on finding the integral with smaller limit $$ I:=\int_{0}^{\frac{\pi}{4}} y \ln (\cos y) d y. $$ As ...
user avatar
  • 5,196
2 votes
3 answers
204 views

Computing $\int_{-1}^{1} \frac{\ln (1+y)}{y} d y$ with simple method.

Using the series $$ \ln (1+y)=\sum_{n=0}^{\infty} \frac{(-1)^{n} y^{n+1}}{n+1} \text { for }|y|<1 $$ to convert the integral into $$ \begin{aligned} \int_{-1}^{1} \frac{\ln (1+y)}{y} d y &=\...
user avatar
  • 5,196
0 votes
1 answer
74 views

Sum of infinite series, Are both of these series equal? 1/2+1/3+1/4...

from an old Numberphile video they explain that the sum of all natural numbers is equal to -1/12, 1+2+3+4+5+...= -1/12. Obviously it diverges, but the -1/12 is meant to be a meaningful representation ...
user avatar
  • 3
0 votes
0 answers
80 views

calculate the integral $\int_{0}^{+\infty} \frac{t^{z-1}\cos(t)}{e^{t}-1} dt$

I am trying to calculate the integral $$\int_{0}^{+\infty} \frac{t^{z-1}\cos(t)}{e^{t}-1} dt$$
user avatar
  • 149
4 votes
1 answer
154 views

$\int_{0}^{1} \frac{ \ln^n x}{1+x^{m}} d x$ =?

Inspired by the post, I try to generalize the result to $$ I(m, n)=\int_{0}^{1} \frac{ \ln ^nx}{1+x^{m}} d x $$ Instead, I first investigate its partner integral $$ \begin{aligned} I(a): &=\int_{0}...
user avatar
  • 5,196
0 votes
3 answers
129 views

How to find the closed form for the integral $I_n:=\int_{0}^{1} \frac{x^{n} \ln x}{1+x^{2}} d x, {} $ where $ n\in N$?

Inspired by the post, I started to investigate the integral in general, $$ I_n:=\int_{0}^{1} \frac{x^{n} \ln x}{1+x^{2}} d x, $$ where $ n\in N.$ Using the infinite geometric series, $$\frac{1}{1+t}=\...
user avatar
  • 5,196
0 votes
1 answer
72 views

The Hurwitz zeta function at the positive integers

Is there a formula that gives the values ​​$\zeta(2n,a)$ as a function of $a$ and Bernoulli numbers, where $n$ is a natural number and $0<a≤1$? $\zeta(z,a)$ is the Hurwitz zeta function.
user avatar
  • 149
-1 votes
1 answer
65 views

Relation for Hurwitz zeta function

I would like to find a proof of the following relation for the Hurwitz zeta function $\zeta(s,a)=\sum_{n=0}^\infty \frac{1}{(n+a)^s}$: $$5\zeta(2,1/3)-\zeta(2,1/6)=\frac{4\pi^2}{3}$$
user avatar
  • 790
0 votes
1 answer
27 views

What is the value of the composite zeta function evaluated at 2?

This is my very first question on this website. I was trying to calculate the value of ζ(4). Although I know the exact value (which I found on google to be pi^4/90) but I wanted to derive it by myself....
user avatar
0 votes
0 answers
94 views

Contour Integral representation Hurwitz Zeta Function over Hankel Contour

I am trying to prove the following contour integral representation of the Hurwitz zeta Function that appears here. $$\zeta(s,a)=\frac{\Gamma(1-s)}{2 \pi i}\int_{H}\frac{ z^{s-1}e^{az}}{1-e^z}\,dz \tag{...
user avatar
  • 2,149
0 votes
1 answer
32 views

proving that dirichlet series has non negative coefficients and does not converge for all $s\in\mathbb{C}$

given $Z(s)=\zeta^2(s)\zeta(s+it)\zeta(s-it)$ I need to prove that Z(s) is represented by a dirichlet series with non negative coefficients whiche does not converge for all $s\in\mathbb{C}$. I have ...
user avatar
4 votes
1 answer
49 views

Relations between different zeta functions for a simple algebra

I'm trying to understand the classical works of Eichler, Shimura and many others (especially Shimizu and Tamagawa's annals papers) on the "classical" (I'm a newcomer and I'm not sure whether ...
user avatar
0 votes
0 answers
51 views

Series for the higher derivatives of the Zeta function around S=1 with Stieltjes constants?

The series for $\zeta'(s)$ about $s=1$ is: $-(\frac1{(s-1)^2}) -\gamma_1 +\gamma_2(s-1)-\frac12\gamma_3(s-1)^2$.... where $\gamma$ are the Stieltjes constants. I'm using this series to determine the ...
user avatar
0 votes
1 answer
72 views

How to plot Riemann zeta function in xy coordinate system? [closed]

Riemann zeta function in complex plane would look like How to plot this curve best as piece wise functions against x-axis in xy plane.
user avatar
  • 113
0 votes
0 answers
46 views

counting primes on different scales than the normal one

Assuming $x-y$ axes on the normal scale we can count primes using $\pi(x).$ We know that a complex function's zeros encode information about the distribution of $\pi(x),$ and that is the Riemann zeta ...
user avatar
  • 733
0 votes
1 answer
57 views

Evaluating $\lim_{n\to\infty}\frac{1}{n+1}(\omega+\nu)^{(n+1)}z^\nu{_2F_1}(1,\omega+\nu+1;n+2;1-z)$

I recently found a proof for the following sum \begin{align*} S_n & =\sum_{k=0}^n\mathcal S_n^{(k)}(\Phi(z,-k,\omega)-z^\nu\Phi(z,-k,\omega+\nu))\\ & =\frac{1}{n+1}(\omega+\nu)^{(n+1)}z^\nu{...
user avatar
1 vote
1 answer
48 views

Does the Zeta Distribution converge to normal as N gets large

I am curious if the zeta distribution converges to normal if it is summed over many times. I am particularly curious if this is true for $\zeta$(4). I know that, if it does converge to normal, it ...
user avatar
  • 81
0 votes
0 answers
40 views

Value of series involving central factorial numbers and zeta function

I encountered an interesing series and was wondering if its value can be computed. First we consider the central factorial numbers. For $n\in \mathbb N$ we define the polynomial $$ P_n(x) = x(x + \...
user avatar
  • 498
1 vote
0 answers
40 views

Difference of polylogarithms of complex conjugate arguments

I have the expression $$\tag{1} \operatorname{Li}_{1/2}(z)-\operatorname{Li}_{1/2}(z^*) $$ Where $\operatorname{Li}$ is the polylogarithm and $^*$ denotes complex conjugation. The expression is ...
user avatar
  • 3,366
0 votes
1 answer
48 views

Properties of the mean residual life function of Zeta distribution

Let's say X is Zeta distributed: $$ p( X = k ) = \frac{k^{-s}}{\sum_{n=1}^\infty k^{-s}} $$ Then, the MRLF (Mean Residual Life Function) of the Zeta distribution is: \begin{align} e(u) &= E[X - u ...
user avatar
  • 103
0 votes
0 answers
97 views

$\sum_{n=1}^{\infty} \frac{\sin(n^2)}{n^2}$

Question: $$\sum_{n=1}^{\infty}\frac{\sin(n^2)}{n^2}=\,?$$ Previously I calculated a similar summation but it was more luck than wisdom, and insight led me to believe my methods were super incorrect (...
user avatar
  • 533
1 vote
2 answers
87 views

$\lim_{t \to \infty}\zeta(\frac{1}{2} + it)$

I am learning about to zeta-function, I am a beginner. I am trying to find: $$\lim_{t \to \infty}\zeta(\frac{1}{2} + it)$$ From the book Theory of the Riemann zeta-function-clarendon by Titchmarsh in ...
user avatar
4 votes
3 answers
231 views

$\frac {\zeta(2)}{e^2} + \frac {\zeta(3)}{e^3} + \frac {\zeta(4)}{e^4} + \frac {\zeta(5)}{e^5}.....?$

$$\frac {\zeta(2)}{e^2} + \frac {\zeta(3)}{e^3} + \frac {\zeta(4)}{e^4} + \frac {\zeta(5)}{e^5}.....?$$ I tried to solve it by using this product formula, $$\frac 1{\Gamma (x)}=xe^{\gamma x} \prod_{n=...
user avatar
0 votes
2 answers
100 views

Prove that $\zeta (s)$ has no essential singularity over $\Re (s) =1$

There are some thoughts just come across my mind about proving $\zeta(s)$ has no poles or essential singularity on $\Re (s) = 1$. So due to the fact that $\zeta(s)$ has an analytic continuation over $\...
user avatar
1 vote
0 answers
29 views

The average value of $\zeta(1/2+it)$ at Gram points is 2

It is stated in $\textit{Riemann’s Zeta Function}$ (H.M. Edwards) in the footnote of p126 that the average value of $\zeta(1/2+it)$ at Gram points is 2 - he cites Section 11.1 of the paper ON VAN DER ...
user avatar
  • 1,144
5 votes
1 answer
43 views

Drinfeld associators expansion to higher weights

Does anyone know where to get higher orders (for weight >10) expansion in Drinfeld associators. (the generating function for MZVs) The answer might be in the form of (4.5) in https://arxiv.org/pdf/...
user avatar
  • 99
1 vote
0 answers
70 views

Counting numbers only containing primes in given set of primes

Suppose $\mathbb P$ is a set of primes, and more specifically I'm interested in infinite $\mathbb P$ but satisfying \[ \sum _{p\in \mathbb P}\frac {1}{p}<\infty .\] How can I evaluate \[ \sum _{n\...
user avatar
  • 1,323
0 votes
0 answers
63 views

Ordinate of zeros of zeta Riemann function

In the book: Multiplicative Number Theory, Davenport, we see that there is a bound for ordinates of zeros of zeta Riemann function ($s=\beta+i\gamma$ is zero of zeta function) My question is that how ...
user avatar
  • 39
3 votes
0 answers
44 views

Proof that $(1-\zeta(1-p,p))/p$ is an integer?

For all primes $p>2$, does the following identity $$\frac{1-\zeta(1-p,p)}{p}\in\mathbb{Z}$$ hold such that $\zeta(s,a)$ denotes the Hurwitz zeta function? If so, how would I go about formulating a ...
user avatar
  • 113
3 votes
1 answer
98 views

Proving an identity for Riemann zeta type sum

I have asked here for a closed form of the sum $$\sum_{n=1}^∞\left(\frac{1}{n^2+\alpha^2}\right)^a.$$ However, it was suggested in the comments that even though a closed form might not be possible, ...
user avatar
2 votes
2 answers
205 views

Euler Maclaurin Formula to prove $\zeta(0)=-\frac12$ and $\zeta(-1)=-\frac{1}{12}$

This is a quick question regarding the analytic continuation of the Riemman Zeta function by application of the Euler Maclaurin Formula and the evaluation of $\zeta(0)=-\frac12$ and $\zeta(-1)=-\frac{...
user avatar
  • 2,149
5 votes
2 answers
170 views

Integer Values of $\sum_{k=1}^n k^r . \sum_{q=1}^n \frac{1}{q^r}$

For harmonic numbers $H_n = \sum_{k=1}^n \frac{1}{k}$ we know that this sum is never integer for any $n$. The same is true for generalized Harmonic numbers: the sum $\sum_{k=1}^n\frac{1}{k^r}$ is ...
user avatar
0 votes
0 answers
51 views

Riemann zeta function zeros notation

In books/papers, we often see the notation that the zeros of the $\zeta$-function with positive imaginary part are denoted $\rho_n$ in order of increasing height. However, because of the equation $\...
user avatar
  • 1,144
3 votes
1 answer
73 views

Prove that $\eta (1/2) \gt \frac{\sqrt 2 \pi^2}{48}$

Prove that $\eta (1/2) \gt \frac{\sqrt 2 \pi^2}{48}$ I am out of any good ideas, but the $\pi^2$ suggests some comparison with $\zeta(2)$. Here is a bad try: $$\eta (1/2) \gt 1-\frac{1}{\sqrt 2} + \...
user avatar
0 votes
0 answers
39 views

Explanation of the proof of absolute convergence of $ \zeta(s, a) $

Here is Theorem 12.1 of Introduction to Analytic Number Theory by Apostol - Theorem 12.1 The series for $ \zeta(s, a) $ converges absolutely for $ \sigma > 1 $. The convergence is uniform in every ...
user avatar
  • 1,132
3 votes
1 answer
122 views

Real part of Riemann Zeta function inequality

In the book ‘Riemann’s Zeta Function’ by H.M Edwards, the following is a line in a proof (within section 6.7) that I can’t follow. The variable $T$ is just a positive real number. $$|\Re [\zeta(2+iT)]|...
user avatar
  • 1,144
2 votes
0 answers
40 views

Epstein zeta function for real quadratic number fields

Let $K$ be a real quadratic number field and $\mathcal O_K$ its ring of integers. Then $$q:K\to \mathbb Q,\quad q(x):=\operatorname{tr}(x^2)/2$$ turns $\mathcal O_K$ into a positive definite lattice ($...
user avatar
0 votes
0 answers
70 views

What is the abscissa of convergence of the series $\sum\limits_{n=1}^{\infty} (-1)^n \frac {1} {n^s}\ $?

What is the abscissa of convergence of the series $\sum\limits_{n=1}^{\infty} (-1)^n \frac {1} {n^s}\ $? In the lecture note our instructor claimed that the abscissa of convergence of the above ...
user avatar
  • 201
1 vote
0 answers
33 views

Dedekind zeta function of a splitting field

Given an irreducible separable polynomial with coefficients in a number field $K$, what, if anything, can one deduce about the Dedekind zeta function of its splitting field only knowing the polynomial?...
user avatar
  • 1,271
2 votes
0 answers
94 views

Computation of $\sum_{n\geq1}(-1)^{n+1}\frac{\text{H}_{n}\zeta(1+n)}{1+n}$ in terms of zeta function

I would like to get an expression which represents the series $$\sum_{n\geq1}(-1)^{n+1}\frac{\text{H}_{n}\zeta(1+n)}{n+1}$$ in terms of $\zeta (2)$ and a series expansion containing $\zeta (n)$. Here, ...
user avatar
  • 5,571
2 votes
1 answer
145 views

On the integral $\int\limits_{0}^{\infty}x^{s-1}\left(\frac{e^{(1-a)x}}{e^{x}-1}-\frac{1}{x}\right)\,dx$

On the integral representation of Hurwitz Zeta Function inside the critical strip, Show that: $$ \Gamma(s)\zeta(s,a)=\int_{0}^{\infty}x^{s-1}\left(\frac{e^{(1-a)x}}{e^{x}-1}-\frac{1}{x}\right)\,dx \...
user avatar
  • 3,537
1 vote
1 answer
82 views

Fast computation of the prime zeta function

What is the state of the art for numerically computing the prime zeta function? There are several papers on the subject for the Riemann zeta function, such as this paper. This paper on the subject ...
user avatar
  • 5,070
3 votes
1 answer
203 views

How do I solve this ominous integral?

Let $ n\ge 1 $ be a positive integer. How do I prove the generalization: $$ \int_0^1\frac{\arctan(x)\log^{2n}(x)}{1+x} \, dx =\frac{\pi}{4}\left(1-2^{-2 n}\right) \zeta(2 n+1)(2 n)!+\frac{1}{2} \beta(...
user avatar
  • 1,788
2 votes
1 answer
152 views

Evaluating $\int_0^y \frac{z^2e^{az}}{e^z-1}\,\mathrm dz$.

Given $a,y\in\mathbb{C}$, I want to find a closed form for $$ \int_0^y \frac{z^2e^{az}}{e^z-1}\,\mathrm dz. $$ Background: This integral was obtained from an infinite sum involving Bernoulli number ...
user avatar
  • 700

1
2 3 4 5
14