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Questions tagged [riemann-zeta]

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

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Riemann Zeta function, nontrivial zeroes

How can we prove what are, say the first 4 non-trivial zeroes of the Riemann $\zeta$ on the critical line $Re(z_j)=\frac{1}{2}$, $j=1,2,3,4$ the first two with negative imaginary part and the second ...
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$\int_{-\infty}^{\infty}\frac{\arctan\sqrt{x^2+2}}{(x^2+1)\sqrt{x^2+2}}dx=\zeta(2)$ Without Feynman Integration

How do I find $$\int_{-\infty}^{\infty}\frac{\arctan\sqrt{x^2+2}}{(x^2+1)\sqrt{x^2+2}}dx=\zeta(2)$$ without Feynman integration? I saw this video, which gives $$\int_{0}^{1}\frac{\arctan\sqrt{x^2+2}}{(...
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Undersanding definition of Riemann-Stieltjes Integral used in Edwards book

I'm trying to find an explaination of the definition of Riemann-Stieltjes Integral used on page 22 of Edwards book [RZ]: This can also be accessed hopefully legally here: Zeta 1) The question is ...
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Using the functional equation of the Zeta function to compute positive integer values

I was reading this article by Ivic. In the introduction, he mentions the functional equation of the Riemann Zeta function, which he says is valid for all complex $s$: $$ \zeta(s)=\chi(s)\zeta(1-s), $$...
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Calculating the $\sum_{x=1}^\infty\left[\prod_{k=0}^s{s\choose k}x^k\right]^{-1}$

Question: If $s \in \mathbb{N}$ is it true: $\sum_{x=1}^\infty\left[\prod_{k=0}^s{s\choose k}x^k\right]^{-1}={\zeta\left[{s+1 \choose 2}\right] \above 1.5pt \prod_{k=1}^s{s \choose k}};$ where $\...
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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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How can I derive this fast-converging series formula for $\zeta(4)$?

Let $\zeta(n)$ denote the Riemann Zeta function for positive integers $n>1$ as usual by: $$ \zeta(n)=\sum_{m=1}^{\infty}m^{-n}. $$ There are fast-converging series for $\zeta(2)$ and $\zeta(3)$, ...
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Why can we not establish the irrationality of Catalan's contant the same was as $\zeta(3)$?

One of the main ingredients in Apéry's proof of the irrationality of $\zeta(3)$ is the existence of the fast-converging series: $$ {\displaystyle {\begin{aligned}\zeta (3)&={\frac {5}{2}}\sum _{k=...
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Identity involving the Möbius function and the first derivative of the Riemann zeta function

Working on the derivatives of the Riemann zeta function, I noted that, for any positive integer $n>1$, the following identity holds: $$\frac{\zeta'(n)}{\zeta^2(n)}=\sum_{x=1}^\infty \mu(x) \frac{\...
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Can the exact number of twin primes $\leq n$ be proved using a “twin-prime zeta function”?

Let $\pi(n)$ denote the amount of primes $\leq n$ and let $\pi_2(n)$ the equivalent for twin primes. Properties of $\pi(n)$ can be proved using a well-known formula involving the zeros of the Riemann ...
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Will this function / symbolic integral converge to the Riemann zeta zero counting function?

The von Mangoldt function can for $n>1$ be computed as: $$\Lambda(n)=\lim\limits_{s \rightarrow 1} \zeta(s)\sum\limits_{d|n} \frac{\mu(d)}{d^{(s-1)}} \tag{1}$$ Formula $(1)$ has been proven by ...
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Proof check of $\zeta(2n)=(-1)^{n+1}\frac{2^{2n-1}B_{2n}\pi^{2n}}{(2n)!}$

Just need to know if my work for this is correct. To start off, I sum over the Laplace Transform of $\sin(ax)$:$$\int_0^\infty\sin(ax)e^{-nx}dx=\frac{a}{n^2+a^2}$$ $$\sum_{n=1}^\infty\int_0^\infty\...
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Questions related to Moebius Transform of Characteristic Function of the Primes

Consider the function defined in (1) below related to the fundamental prime counting function $\pi(x)$. Note that A143519(n) is not multiplicative. (1) $\quad f(x)=\sum\limits_{n=1}^{x}A143519(n)$ ...
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Zeta infinity product

I have been trying to calculate this: the product of Zeta function (n) when n=2 to infinity. It is a convergent product that approaches 2.295, but can the result be arrived to analytically.
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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 ...
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Deletion of Pole of functional equation for zeta function

$\Phi (s) = \Phi(1-s) = \pi ^ {-s/2} \Gamma(\frac{s}{2}) \zeta(s) = \pi ^ {-(1-s)/2} \Gamma(\frac{1-s}{2}) \zeta(1-s)$ It has 2 simple pole for $s=0,1$ Now it derives another function deleting poles ...
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Show that $\frac3{16}\sum_{n=1}^{\infty}\frac{(-1)^n-8}{n^3}=-\frac{105}{64}\zeta(3)$

Consider the infinite sum $$\frac34\sum_{n=1}^{\infty}\left[\frac{(-1)^n}{(2n)^3}-\frac2{n^3}\right]=\frac3{16}\sum_{n=1}^{\infty}\frac{(-1)^n-8}{n^3}$$ I have come across this sum while ...
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Does $\zeta(2n+1)/\sqrt{\zeta(2(2n+1))}$ converge for $n\to\infty$?

I know convergence questions involving the Riemann $\zeta$-function at odd positive integers are hard to answer. However, I came across the following on internet, but forgot the website/blog that it ...
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$\sum_{n=1}^{4000000} \frac{1}{n^3}$ very quick.

Some days ago I have tried to find the sum of the first milion terms of the infinite sum $\zeta(3) = \sum_{n=1}^\infty\frac{1}{n^3}$ (Apéry's constant) on Wolfram Programming Lab (Open Cloud), an ...
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Theorem 14.25(A) in Titchmarsh “The theory of the Riemann zeta-function”

In Titchmarsh's book "The theory of the Riemann zeta-function" there's theorem 14.25(A) on page 369 of the second edition where a summand $1/\zeta(s)$ appears out of the blue, so it seems... Oh, I do ...
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Is it possible to plug hypercomplex numbers into the Riemann Zeta function?

I'm aware of the more detailed question: How to raise a number to a quaternion power However, from a more high-level perspective (read: probably less mathematical, hence my choice for raising ...
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Laurent Series of the Log-derivative of the Xi Function

So I am curious about the product representation of the Xi function. Some operations applied to the Hadamard product can result in the identity: $$\frac{\xi'(x) }{\xi(x) }=-\sum_{n=0}^{\infty }x^n\...
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What is the mass center of the Riemann Zeta Function across the critical line?

I just came with the idea: what is the center of mass of the Riemann Zeta Function across the critical line? I mean: when you plot the parametric graph across the critical line, you get the famous ...
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Modified Riemann Zeta function - where do the non trivial zeros live now?

I'm curious as to where the non trivial zeros of the Riemann Zeta function live after you modify it in the following way: Take the multivariate complex function: $\phi(s,t)=\zeta(t)^{log(s)}, $ ...
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1answer
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Is it true that $\sum\limits_{n=1}^\infty\frac{\chi_{k,1}(n)}{n^s}=\zeta(s)\sum\limits_{d|k}\mu(d)\,d^{-s}$?

Question 1: Is the relationship illustrated in (1) below true where $\chi_{k,1}(n)$ corresponds to the ordering of Dirichlet characters implemented by Mathematica? (1) $\quad\sum\limits_{n=1}^\infty\...
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What makes proving the Riemann Hypothesis so difficult?

iam a Biology major (please don't shame me!) but i really enjoy mathematics. Recently i have been reading about this conjecture and its importance in understanding the distribution of primes. After ...
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Use for $ \zeta’’(s) $ in number theory?

Let $\zeta(s) $ be the Riemann zeta function. Let $\zeta’(s) $ and $\zeta’’(s) $ be the first and second derivative of that Riemann zeta function. In analytic number theory I see the use of $\zeta(s) ...
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Convergence of Euler Product on the line Re(s) = 1

Consider the Euler product for the Riemann Zeta function: \begin{equation} \zeta(s) =? \prod_p (1-p^{-s})^{-1} \end{equation} When I started studying this product, I read that for $Re(s) > 1$, the ...
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An Integral Representation of Logarithmic Derivative of Zeta Function

How to show that: $$ \int_{0}^{\infty}\frac{1}{e^x-1}\left[\frac{12\,e^x}{(e^x-1)^2}-\frac{12}{x^2}+1\right]\,dx=\frac{5}{2}+\frac{\zeta'(2)}{\zeta(2)}-\frac{\zeta'(0)}{\zeta(0)}\tag{1} $$ Is it ...
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Is there an analytic function with zeroes only at $-2n$, and zeroes at $\frac12\pm it$, and further, symmetric zeroes within the critical strip?

Is there an analytic function with zeroes only at: every $-2n$, $\frac12\pm it$, and at least one at $\frac12\pm\epsilon\pm it$ where $0<\epsilon<\frac12, t\neq0$ (and these zeroes observing ...
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regularizing function under sqrt [closed]

Is it possible to find finite part of $\Sigma^\infty_n\sqrt{n^2 +a^2}$ using something like regularized zeta function?
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Domain coloring of Riemann zeta function

This is the domain coloring of Riemann zeta function. I do get the Re(z) < 1 part of the graph. However I am unsure why the Re(z) > 1 part of the graph is all red with the same brightness and hue. ...
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Is my interpretation of formula correct?

I am trying to confirm Littlewoods's Theorem/criteria for Riemann's hypothesis which is based on Mertens' function, $M(x)$, by developing a formula for $M(x)$ as $x \to \infty$: Theorem-Littlewood ...
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Zeta function generalized to quaternions?

Has the $\zeta(s)$ function, $\sum_n 1/n^s$, been generalized to quaternions, so $\zeta(q)$ for $q$ a quaternion? Euler defined it for $s$ integers, Chebyshev for $s$ real, Riemann for $s$ complex. ...
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Disjoint sets in a combinatoral sum (continued)

Let $f_S(m)$ be defined as in my previous question: Let $S = \{1/n^2 : n \in \mathbb{N} \}$. Let $f_S(m)$ be the sum of the products of all $m$-tuples chosen from $S$. That is $$f_S(m) = \sum_{...
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Is it possible to isolate and thereby by integration compute a zeta zero gap when accentuating the zeros and counting them?

The starting point is this integrable formula for the von Mangoldt function: $$\Lambda(n)=\lim\limits_{s \rightarrow 1} \zeta(s)\sum\limits_{d|n} \frac{\mu(d)}{d^{(s-1)}}$$ We can plot the Dirichlet ...
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Sum of products of $m$-tuples chosen from the set of squared reciprocals

Let $S = \{1/n^2 : n \in \mathbb{N} \}$. We know $\sum S = \zeta(2) = \pi^2/ 6$. Let $f(S, m)$ be the sum of the products of all $m$-tuples chosen from $S$. That is $$f(S,m) = \sum_{X \in {S \...
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Laurent series of $\zeta$ function around $\infty$?

My question essentially is: What is the Laurent series of $\zeta$ function around $\infty$? I know that this can be done by finding the Laurent series of $\zeta(\frac1z)$ around $0$. For the ...
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Does there exist positive rational $s$ for which $\zeta(s)$ is a positive integer?

Does there exist positive rational $s$ for which the Riemann Zeta function $\zeta(s) \in N$ or equivalently, does there exist finite positive integers $\ell,m$ and $n$ such that $$\zeta\left(1+\dfrac{\...
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3answers
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Does a generating function for $\zeta(2k+1)$ exist?

I know that a generating function for the Zeta function at the even integers already exists, but how about the Zeta function at the odd integers? I've done some research, and found some alternative ...
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Calculating the first zero of Riemann Zeta Function by hand

There are lots of pages on MSE and other websites regarding few first zeros of The Riemann Zeta Function. On MSE [this] is by far the closest (or the only one I've found) explanation for finding the ...
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Can we prove $B (n) = \frac{1}{4} G (n - 1) G (n)$ is an indicator function that takes on the value 1 for 'bad' and 0 for “good” Gram points?

Let \begin{equation} g (n) = 2 \pi e^{1 + W \left( \frac{8 n + 1}{8 e} \right)} \end{equation} be the approximate value of the $n$-th Gram point. Let \begin{equation} G (n) = \frac{Z (g (n))}{...
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How is this equation obtained? $\int_0^\infty \frac{x^{s-1} dx}{e^x-1}=\Pi(s-1)\zeta(s) $ [duplicate]

This equation: $$\int_0^\infty \frac{x^{s-1}}{e^x-1} dx=\Pi(s-1)\zeta(s) $$ was used by Riemann in his famous paper from 1859. Seemingly it follows from: $$ \int_0^\infty e^{-nx}x^{s-1}dx= \frac{\Pi(s-...
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Notation: What does “$p-1|n$” mean in “$\prod_{p-1|n} p$”?

I'll try my best to reconstruct how this appears in my pdf: $$d:denom(B_n)=\prod_{p-1|n} p$$ the "$p-1|n$" part, I don't understand. I get this has something to do with Riemann Zeta Function, and ...
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Mathematical Explanation of Mathematica Summation ${\sum_{n=1}^{\infty}\frac{(2n-1)!}{(2n+2)!}\zeta(2n)}$

From a mathematical point of view, what phenomena that most likely Mathematica Wolfram encountered when calculating: $$ \sum_{n=1}^{\infty}\frac{(2n-1)!}{(2n+2)!}\zeta(2n)\,=\,\color{red}{\frac{2\...
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1answer
53 views

Is there a zeta function(with a Dirichlet series) having known roots off the critical line?

Is there a zeta function(with a Dirichlet series) having known roots off the critical line? I thought there was something like the Hilldebrand-Davis zeta function or something like that, but I can't ...
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210 views

What is $f(2s+1)$ when $f(s)=\sum_{n=0}^\infty {\frac{(-1)^n}{(2n+1)^s}}=1-\frac{1}{3^s}+\frac{1}{5^s}-\frac{1}{7^s}+\dots$? [duplicate]

Is there an exact form of $$f(s)=\sum_{n=0}^\infty {\frac{(-1)^n}{(2n+1)^s}}=1-\frac{1}{3^s}+\frac{1}{5^s}-\frac{1}{7^s}+\dots$$ when $s$ is odd? Discussion I have been exploring infinite series ...
3
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1answer
96 views

Is there another way of evaluating $\lim_{x \to 0} \Gamma(x)(\gamma+\psi(1+x))=\frac{\pi^2}{6}$

I was messing around with the Zeta-Function and I got what I thought was an interesting limit: $$\lim_{x \to 0} \Gamma(x)(\gamma+\psi(1+x)) = \frac{\pi^2}{6} $$ Where $\Gamma$ is the gamma function, $...
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1answer
110 views

Motivation for the Basel problem

I realized that I know of several ways how to prove that $\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$, but I have no idea why I would want to know the answer in the first place. Answers I have ...
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1answer
58 views

Calculating the symbolic solution to a definite integral containing trigonometric functions

It is verifiable numerically or with a computer algebra system (for example with Mathematica using NIntegrate) that the numerical solution to the following integral ...