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Questions tagged [riemann-zeta]

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

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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
24 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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26 views

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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77 views

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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51 views

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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Stein's analytical continuation of Riemann's zeta function

This is a problem in the book by Stein and Shakarchi. I am puzzled. The first integral indicates that at s=1−m=1,0,−1,−2,...., there is a pole. How did them come to the conclusion that the ...
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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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1answer
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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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240 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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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(...
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2answers
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Need reference for proof of functional equation for generalized L-functions

When proving functional equation for Riemann zeta function one starts at the definition of gamma function $$\Gamma(s) = \int_0^{\infty} x^{s-1} e^x\mathrm dx\tag1$$ After a few steps we arrive at $$ ...
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Closed form solution for $\int_0^1 \frac{\ln^2(x)\ln^2(1-x)}{x(1-x)} dx$ using Bose Integral

I'm looking for the solution to the following integral, but by using the Bose integral: $$\int_0^1 \frac{\ln^2(x)\ln^2(1-x)}{x(1-x)} dx$$ I got to this form when looking for a solution to the ...
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1answer
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What is the Dirichlet Transform of $a(n)=\frac{\mu(n)}{n^2}\,\log\left(\frac{2\,\pi}{n}\right)$?

This question is related to my previous question at the following link. Do the Prime Number Theorem and/or Riemann Hypothesis predict a limit on the accuracy of this formula for $\gamma$? This ...
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1answer
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Conditional convergence of series $\sum_{n=1}^{\infty}a_n(\frac{k}{n}-k^{n+1}\frac{\zeta(n+1)}{n+1}),0<k\le 1$

Does the series $\displaystyle \sum_{n=1}^{\infty} a_n\left(\frac{k}{n}-k^{n+1}\frac{\zeta(n+1)}{n+1}\right),0<k\le 1$ converges conditionally given that $\sum_{n=1}^{\infty}|a_n|<\infty$. ...
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51 views

Partial Euler product

The Riemann Zeta function defined as $$\zeta(s) = \sum_{n=1}^{\infty} n^{-s}$$ For $\Re(s)>1$ is convergent and admits the Euler product representation $$\zeta(s) = \prod_p (1-p^{-s})^{-1}$$ For ...
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Riemann zeta partial sums

Playing with the Riemann zeta function I came across a property of partial sums of $\zeta(s)$ that I wonder if anyone could explain. Taking a partial sum as$$\zeta_{ab}(s)=\sum_{n=a}^b\frac{1}{n^s}$$ ...
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Counting numbers of Fixed point of Zeta function by Argument Principle

This is my first post about this topic and now I am trying to evaluate the integral, $$N=\frac{1}{2\pi i}\oint_{|z-1|=1}\frac{\zeta'(z)-1}{\zeta(z)-z}dz+1$$ $\zeta-$is the Riemann Zeta function. I am ...
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Is the Dedekind zeta function always non-vanishing at $s=\frac{1}{2}$

Is the Dedekind zeta function of a number field always non-zero at $s=\frac{1}{2}$? For Riemann zeta function, it's true by direct computation. If this is true for quadratic fields, then one can use ...
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Can the value of the Riemann zeta function at $n=2$ be derived from the Wallis formula for $\pi$?

It is well known that the Riemann zeta function, defined for all positive integers $n>1$ by $$ \zeta(n) = \sum_{m=1}^{\infty} m^{-n} $$ takes the value $\displaystyle \frac{\pi^2}{6}$ at $n=2$. On ...
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Sum of almost-prime zeta functions

Wiki defines an almost-prime zeta function as a sum of inverse powers of the k-primes (the integers which are a products of $k$ not necessarily distinct primes): $$P_k(s)=\sum_{n: \Omega(n)=k} \frac{...
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51 views

Riemann Zeta function, perfect powers, and the Mobius function

I was toying around with the Riemann Zeta function recently and noticed that I could get to a particular representation (valid for $Re(s)>1$) in a couple of different odd ways. The first was by ...
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What information do the moments of the Riemann-Zeta function give us

I have seen an explicit formula for what a moment of the Riemann-Zeta function is but I am unsure what information this give us? If we are looking at the zero's of the function then this can be ...
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Probability density function for Riemann-zeta zeros

Curious about the expected probability distribution for the spacing between Riemann zeta zeros, of the form $s_n=\sigma+it_n$, where $\sigma=0.5$ and $t_n$ is the imaginary part of the $n$-th zero. ...
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1answer
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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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Why does the following not show $\zeta(0) = -\frac{1}{2}$?

I am trying to evaluate the $\zeta(s)$ at $s=0$, but I am not sure what is incorrect about the following? \begin{equation*} \label{eq:RiemannzetaFinal} \zeta(s) = 2^s\pi^{s-1}\sin \Bigl(\frac{s\...
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On $\sum_{k=1}^\infty1/(k!k^s)$.

Has the following $\zeta$-like function been studied before? $$f(z;s)=\sum_{k=1}^\infty\frac{z^k}{k!k^s}.$$ I believe this is an entire function since using the ratio test, $$\lim_{n\to\infty}\frac{...
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von Mangoldt's formula for Chebyshev $\psi$ function

Chebyshev's $\psi$ function is defined for primes $p$ as $$\psi(x)=\sum _{p^k\leq x} \log (p)$$ von Mangoldt found an explicit formula for this, with the exception that the function takes half-...
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Obscure approximate functional equation for the Riemann zeta function

The following result is supposed to follow from an approximate functional equation for the Riemann zeta function, but I've never seen anything close enough to it : For $T \leq t \leq 2T$ where $T$ is ...
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51 views

Approximation of $\zeta$(s) at $s=1$

I am currently taking an Analytic Number Theory unit and we're working on the zeros of the zeta function. In the proof of $\zeta(1+\textit{i}t) \neq 0$ for $t \in \mathbb{R}$, we suppose that $\zeta$(...
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Graphical representation of the zeta function in the region $0 < s <1$ for real $s$

Riemann zeta function can be defined by the equation $$ \zeta(s)= \lim _{x\to \infty} \left( \sum_{n\le x} \frac {1} {n^{s}}-\frac{x^{1-s}} {1-s}\right)$$ if $0<s<1$ (see for example Tom M. ...
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1answer
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How can I know if something is publishable? [closed]

I have found an approximation for $\zeta(3)$ The result includes a numerical value and an infinite series which I can also give as an infinite product. The actual value of $\zeta(3)$ and my ...
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1answer
102 views

Integral $\int_0^1 x^n\left\{\frac{k}{x}\right\}dx$

I am trying to solve the following integral containing fractional part function (denoted by $\{.\}$) $$\int_0^1 x^n\left\{\frac{k}{x}\right\}dx,\ 0<k\le 1,\ n\in \mathrm N^*$$ For $n=0$, it is ...
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1answer
78 views

Convergence of Riemann spectrum/Fourier transform of prime powers

Prime Numbers and the Riemann Hypothesis by Mazur and Stein makes use of an interesting function: $$\hat{\Phi}_{\le C}(\theta)=2\sum_{prime\:powers\:p^n\le C}p^{-n/2}\cdot log(p)\cdot cos(n\cdot log(p)...
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2answers
235 views

Could this problem be similar to the Riemann Hypothesis?

I've found the below equivalence. For $a,b,x\in\mathbb{C}$, provided that there are no singularities on the right-hand side: \begin{multline}\sum _{k=2}^{\infty}\sum _{j=1}^{\infty}\frac{x^k}{(a ...
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1answer
64 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
102 views

Question related to derived formula for zeta-zero counting function

This question assumes the definitions of the Mellin and Fourier transforms illusrated in (1) and (2) below and the corresponding relationship illustrated in (3) below. (1) $\quad\mathcal{M}_x[f(x)](s)...
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1answer
56 views

Riemann-zeta Function Evaluated at $\zeta(0)$ [duplicate]

WolframAlpha says that $\zeta(0) = - \frac{1}{2}$ but I can't seem to get that result. I found that for $\Re(s) < 1 $, \begin{equation}\label{1} \zeta(s) = 2^s \pi^{s-1}\sin\Bigl(\frac{s\pi}{2}\...