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Questions tagged [riemann-zeta]

Questions on the famed $\zeta(s)$ function of Riemann, and its properties.

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Can one compute $\zeta_{\mathbb{Z} \times G}(s)$?

For a possibly infinte group $G$, Marcus du Sautoy and Fritz Grunewald define: $$\zeta_G(s) = \sum_{n=1}^\infty \frac{a(G,n)}{n^s} = \sum_{ H \le_f G} \frac{1}{[G:H]^s}$$ where $a(G,n) = |\{H \le G : ...
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RH & explicit formula for the number of primes $ \le x$

Does the RH have to be true in order for Riemann's explicit formula for the number of primes <= x to hold? In other words, does this formula hold even if not all the critical roots are on the ...
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What can be said about $\prod_{s=2}^{\infty} \zeta(s) $?

Another problem from quora. What can be said about $v =\prod_{s=2}^{\infty} \zeta(s) $? Wolfy says that $v \approx 2.294856591673313794183 $. The Inverse Symbolic Calculator (http://wayback.cecm....
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Double harmonic series $\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\frac{H_{n+m}^{(p)}}{(n+1)^{q}(m+1)^{r}}$

Do these sums exist in the literature and have been investigated before? The same question for the odd variant, that is $$ \sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\frac{O_{n+m}^{(p)}}{(2n+1)^{q}(2m+1)^{...
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Is Mertens function bounded?

Let $$M(N)=\sum_{n=1}^{N}\mu(n)$$ be a Mertens function. According to the fact that $$\frac{1}{\zeta(s)}=\sum_{n=1}^{\infty}\frac{\mu(n)}{n^{s}}$$ And fact that $\zeta(0)=-\frac{1}{2}$ Then i ...
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108 views

Integrals of the form $\int \log(2+2\cos x)^ndx$

$\log$ will be the natural logarithm and $\zeta$ the Riemann zeta function. I'm interested in the following family of integrals: $$ I_n = \int_0^\pi(\log(2+2\cos x))^n\mathrm{d}x $$ Some of the values ...
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Why probability of picking a random prime is 0? [duplicate]

"It's well known that there are infinitely many prime numbers, but they become rare, even by the time you get to the 100s," Ono explains. "In fact, out of the first 100,000 numbers, only 9,592 are ...
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Unit Circle mapped to riemann zeta function equation [on hold]

I've numerically run the unit circle looking like this: through python's mpmath's zeta function and got something looking like this: At first I thought it might be gaussian, but I couldn't really ...
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Complex estimate

I´m need some help with the last part of that estimate Let $s_k :=1+\frac{2\pi i k}{\log(2)}, k \neq 0$. Then: $\sum\limits_{n\leq x}(-1)^{n}n^{-s_k}=2^{1-s_k}\sum\limits_{n\leq x/2}n^{-s_k}-\sum\...
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Unit cirle through Riemann Zeta Function

I see that the unit circle gets mapped to something that looks vaguely gaussian but scaled and rotated by $\pi/2$. So I see that if you plug in $e^{ix}$ to the function, you get a sum that can be ...
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Confusion with the Riemann Zeta function

The Riemann Zeta function $\zeta(s)$ satisfies the functional equation $$ \zeta(s)=2^s\pi^{s-1}\sin\left(\frac{\pi s}{2}\right)\Gamma(1-s)\zeta(1-s) $$Because of this, it is obvious ("trivial") that ...
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Why is this sum involving the Riemann Zeta function convergent for $\Re(s)>-1$?

In this Mathworld article, the following function is defined (equation $[12]$): $$ F(s) = (1-2^{1-s})\zeta(s), $$ where $\zeta(s)$ is the Riemann Zeta function defined by the usual infinite sum ...
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What is the density of integers which can be written as the sum of two positive integer powers?

The power-free natural (asymptotic) density of $x^k$ is $1/\zeta(k)$ plus error terms. I suppose this means that the number of integers less than $N$ that a have $k$ power in their factorization is: ...
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Integral $\int_{-\infty}^{\infty}\ln(2-2\cos(x^2))dx=-\sqrt{2\pi}\zeta(3/2)$

Prove that $$\int_{-\infty}^{\infty}\ln(2-2\cos(x^2))dx=-\sqrt{2\pi}\zeta(3/2)$$ I was given this integral in my post Request for crazy integrals. I have never seen an integral like this before ...
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Upper bound of Riemann Zeta function

In the book of Apostol Introduction to Analitic number theory at pag. 271 about $\zeta$ upper bound $\sum_{n=1}^{N} \frac{1}{(n+a)^{\sigma}} \leq 1 + \int_{1}^{N}\frac{dx}{(x+a)^\sigma} $ with $0<...
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Closed form for the sum?

Is there a closed form for this sum? It's a mixing summation of different terms in the zeta function with different values of $s.$ $$ S=\frac{1}{1^2}+\frac{1}{2^3}+\frac{1}{3^4}+\frac{1}{4^5}+ \cdot\...
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Is this zeta related function surjective? [closed]

Let $\psi:[1,\infty[\to ]0,\infty[$ defined as $\psi(t)=|\zeta(1+it)|$, $t\geq 1$. Is $\psi$ surjective? Or there exist $t_0>0$ and $t_1>1$ for which $\psi:[t_1,\infty[\to ]t_0,\infty[$ is ...
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Integration of $\xi$ function over rectangle

Hey simple question (I hope so), I´m reading Introduction to Analytic and Probabilistic Number Theory, and I don't get this step. The author writes, this equation works by a "well known formula". ...
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Elementary approximations to $\zeta(s)?$

What are the best approximations in terms of elementary functions of one real variable for: $$\zeta(s)=\sum_{n=1}^{\infty} \frac{1}{n^s},$$ for $Re(s)>1?$ There is not an elementary function that ...
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On this integral $\int_0^1 \ln\left(\zeta(s)\right)ds$ [duplicate]

I am interested in the integral over the log-zeta function$$I=\int_0^1 \ln\left(\zeta(s)\right)ds$$ My attempts: Sub $u=1-s$ to get $$I=\int_0^1 \ln\left(\zeta(1-u)\right)du=\int_0^1 \ln(\zeta(u))-u\...
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On binomial sums $\sum_{n=1}^\infty \frac{1}{n^k\,\binom {2n}n}$ and log sine integrals

Seven years ago, I asked about closed-forms for the binomial sum $$\sum_{n=1}^\infty \frac{1}{n^k\,\binom {2n}n}$$ Some alternative results have been made. Up to a certain $k$, it seems it can be ...
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Real values of $\frac{\zeta(2 s)}{\zeta(s)}$

If $\frac{\zeta(2 s)}{\zeta(s)}$ is a real number, then must $s$ be real ?
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asymptotic behaviour of $\vert \frac{\zeta(s)}{\zeta(1-s)}\vert$

I have to show with "a simple calculation" that $\vert\frac{\zeta(s)}{\zeta(1-s)}\vert =\vert 2(2\pi)^{s-1}\Gamma(1-s)\sin(\frac{1}{2}\pi s)\vert \sim (\frac{\vert\tau \vert}{2\pi})^{\frac{1}{2}-\...
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Are these formulas for the Riemann zeta function $\zeta(s)$ globally convergent?

This question assumes the following definitions. (1) $\quad S(x)=x-\left(\frac{1}{2}-\frac{1}{\pi}\sum\limits_{k=1}^f\frac{\sin(2\,\pi\,k\,x)}{k}\right),\quad f\to\infty$ (2) $\quad S'(x)=1+2\sum\...
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Deriving Analytic Continuation of Riemann Zeta Function [duplicate]

I'm trying to derive the formula for analytic continuation of riemann zeta function but I can't find a way. How do we find results like Zeta(-1)= -1/12 ?
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Integral estimate for Hankle´s Contour

I have to proof the following estimate $\vert \int\limits_{H_k}z^{s-1}(e^{z}-1)^{-1}dz\vert \leq k^{\sigma}$ Where $H_k$ ist the Hankel Contour with radius $\rho_k = (2k+1) \pi$ From another ...
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Riemann Zeta function with the prime counting function in place of $n$

Interested in the following function: $$ \Psi(s)=\sum_{n=2}^\infty \frac{1}{\pi(n)^s}=\sum_{n=1}^\infty \frac{\lambda_n}{n^s}, $$ where $\pi(n)$ is the prime counting function. Was thinking about: ...
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1answer
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Is this infinite product formula for $\zeta(2)$ interesting?

I can prove that $$ \zeta(2) = \frac{\sqrt6}{2} \prod_{n\geq1} \frac{3n3n}{(3n+1)(3n-1)}\frac{4n4n}{(4n+1)(4n-1)}, $$ where $\zeta(k)$ denotes the Riemann zeta function. I also have other similar ...
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Name of Hardy Littlewood Lemma/Result

I am interested in finding the variation of argument of a certain analytic function $g(z)$ on a region of the form $\frac{1}{2}\leq \sigma \leq \sigma_1$, $t_0\leq t \leq T$ where $\sigma$ stands as ...
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1answer
32 views

Hurwitz zeta function for $s=0$ $\zeta(0,1/2)$

I'm studying the Casimir Effect with perfect spherical boundary which involves the use of the Hurwitz zeta function. I've been staring for a while at this equation: \begin{align} \sum_{l=1}^{\...
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How to prove that Re(zeta(1+i/n)) is equal to the euler constant as n -> infinity

I have noticed that $\displaystyle\lim_{n \to \infty} \Re\left[\zeta\left(1 + \frac{i}{n}\right)\right] = \gamma$ How can this be proven?
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The isomorphism of $\mathbb{Z}_{p}[x] /\left(x^{n}-1\right)$ using Hensel's Lemma

I an trying to prove the following.Let p:prime, $n\in\mathbb{N}$ with $(n,p)=1$ ,and $x^{n}-1=f_{1}(x) \cdot \ldots \cdot f_{r}(x) \quad\left(f_{1}(x), \dots, f_{r}(x) \in \mathbb{Z}_{p}[x]\right.$ ...
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1answer
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Ratio of $\zeta(s)/\zeta(1-s)$ in the critical strip

Question about the Riemann zeta functional equation: $\zeta(s) = 2^s \pi^{s-1}sin(\frac{\pi s}{2})\Gamma(1-s)\zeta(1-s)$ $s=\sigma+it$ Taking $f(s)=2^s \pi^{s-1}sin(\frac{\pi s}{2})\Gamma(1-s)$, ...
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Question about Dirichlet Series Related to Formula for $\frac{1}{e}$

This question is related to the three functions defined in (1) to (3) below where $\coth(z)$ gives the hyperbolic cotangent of $z$. (1) $\quad M(x)=\sum\limits_{n=1}^x\mu(n)\quad\text{(Mertens ...
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Is $\zeta(s)\sim\sqrt{\frac{\zeta(4s)}{\zeta(2s)}}\prod\limits_{n=1}^\infty\big(1-\frac{2}{p_n^s+p_n^{-s}}\big)^{-1/2}$?

The Riemann Zeta function, denoted by $\zeta(\cdot)$, is defined by the following equation for $s > 1$ and $p_n$ the $n^\text{th}$ prime number. $$\zeta(s)=\prod_{n=1}^\infty\bigg(1-\frac{1}{p_n^s}\...
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Question on convergence of formula for Dirichlet eta function $\eta(s)$

The Dirichlet eta function $\eta(s)$ is related to the Riemann zeta function $\zeta(s)$ as illustrated in (1) below. References (1) and (2) claim formula (2) for $\zeta(s)$ is globally convergent (...
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Orientation in the complex plane of $\zeta(s)$ and $\zeta(1-s)$ near a zero

A simple observation of the behavior of $\zeta(s), s=\sigma +it$ that I wonder if there's a explanation for: Take $t_k$ as the height of the $kth$ non-trivial zero $z_k$ on the critical line ($\sigma=...
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Binomial expansion with gamma function

I'm studying Euler proof of $\zeta(2)$. There are several proof but I'm on one. I found in a book this expression for binomial but I can't find where it comes from $(1+y)^{-t} = \frac{1}{\Gamma(t)} \...
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How can i prove the following Infinite Product?

I understand how to derive the Euler Product over primes for the Zeta function using the sieving method. However, upon reading more it said the following.. $$\prod_{P Prime}^{\infty}(1-p^{-s})^{-1} = ...
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Ratio of $\zeta(s)$ and $\zeta(1-s)$ in the functional equation

Question about the $\zeta$ function and the functional equation: $\zeta(s) = 2^s \pi^{s-1}sin(\frac{\pi s}{2})\Gamma(1-s)\zeta(1-s)$ Taking $f(s)=2^s \pi^{s-1}sin(\frac{\pi s}{2})\Gamma(1-s)$, then ...
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Sum of almost-prime zeta functions. Part II

I've already asked a question regarding the Sum of almost-prime zeta functions. Now I'm interested in the next question, denote: $$\zeta_{k}^{al}(s, N)= \sum_{n=1}^{N} \frac{a(n)}{n^s},$$ where $$a_k(...
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Summation of Integrals without using $\zeta(2)$

How do you evaluate the sum $$\sum_{k=1}^\infty\int_{\sqrt k}^{\sqrt{k+1}} \left(\frac{x^2}{k}-1\right)\, dx$$ without using $\zeta(2)$? I can see some relationship to $\dfrac1{\lfloor x^2\rfloor}$ ...
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1answer
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evaluating a double integer series

Consider the series $$\sum_{m,n=1}^{\infty}\frac{f(m,n)}{m^2n^2}$$. In addition, assume that the function $f$ is weakly multiplicative. Then, how could we evaluate the sum. In particular how can we ...
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What is $\zeta(i \infty )$ if $ \zeta(\infty)=1 $ and what is its geometric interpretation?

If we want to compute $ \zeta(\infty) $ for large enough real number we can get $1$ as claimed by Wolfram alpha here which means $ \lim_{s\to \infty} \zeta(s)$ with $s$ is a complex number with nul ...
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Differentiability of the Riemann zeta function [duplicate]

Try to prove that $$ \zeta (s) = \sum\limits_{n=1}^{\infty}\frac{1}{n^{s}}$$ is continuously differentiable when $s\in \mathbb{R}$ and $s>1$. Because $s\in \mathbb{R}$,can I take it just as series ...
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270 views

A series related to prime numbers

Let $H(t) = \sum_{n=1} ^{\infty} \pi(n)t^n$ where $\pi(n)$ is the prime counting function. This is the Hilbert series of some $\mathbb{Q}$-vector space. By the prime number theorem, the radius of ...
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2answers
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Evaluating $\int\limits_0^\infty \mathcal{e}^{-x}\ln^{2}x\,dx$

I'm looking for a proof of this: $$\int\limits_0^\infty \mathcal{e}^{-x}\ln^{2}x\,dx = \gamma^{2}+\frac{\pi^{2}}{6}$$ My first thought would have been to write $e^{-x}$ as an infinite series and ...
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The use of limit in Titchmarsh's book “The theory of the Riemann zeta-function” in Theorem $3.13$

In Titchmarsh's book "The theory of the Riemann Zeta-function" his Lemma $3.12$ is one of the main tools. Lemma $3.12$ is a version of Perron formula. Lemma $3.12$ starts by observing that, with $c>...
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For what real part of $s$ as a function of $q$ is the Euler-Maclaurin formula a valid analytic continuation of the Riemann zeta function?

The familiar formula for the Riemann zeta function: $$\zeta(s)=\lim\limits_{k \rightarrow \infty} \left( \sum\limits_{n=1}^{n=k} \frac{1}{n^s}\right) \mbox{ is true for } \Re(s)>1$$ adding one ...
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Is this $\zeta(p)=\sum_{k\geq1 ,p\in \mathbb{P}}\frac{1}{{\sigma}_k(p)-1}$ true with sigma is power of sum divisor function?

let ${\sigma}_k(n) =\sum_{d|n} d^k$ is a sum of divisor function , And let ${\sigma}_k(n)$ be the iterating divisor function, We have for every prime $p$ and for every integer $k\geq 1$ : $${\sigma}_k(...