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Questions tagged [riemann-zeta]

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

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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 finite positive integers $\ell,m$ and $n$ such that $\zeta(1+\ell/m) = n$?

Does there exist finite positive integers $\ell,m$ and $n$ such that $$\zeta\left(1+\dfrac{\ell}{m}\right) = n$$ I have been trying to solve this question with success. Computationally I have checked ...
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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' 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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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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Can this function's positivity $\forall t \gt 5.56..$ be criteria for the Riemann hypothesis?

Let $Z \left( t \right) ={{\rm e}^{i \left( -i/2 \left( {\it ln\Gamma} \left( 1/4+i/2t \right) -{\it ln \Gamma} \left( 1/4-i/2t \right) \right) -1/2\,\ln \left( \pi \right) t \right) }}\zeta \...
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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 ...
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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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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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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 ...
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Prime gaps and gaps between successive critical zeros of zeta

Assuming RH, the sequence of critical zeros of the Riemann zeta function can be viewed as the Fourier transform of the sequence of primes. From a physicist point of view, the average gap between the ...
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Show that $\int_0^1 4 \space\operatorname{li}(x)^3 \space (x-1) \space x^{-3} dx = \zeta(3) $

My mentor tommy1729 wrote $\int_0^1 4 \space \operatorname{li}(x)^3 \space (x-1) \space x^{-3} dx = \zeta(3) $ I wanted to prove it thus I looked at some methods for computing integrals and also ...
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What is the geometric interpretation of this limit :$\lim_{n\to {+\infty}}\zeta(n)+\zeta(\frac1n)=\frac12$?

it is easy to show that $\lim_{n\to {+\infty}}\zeta(n)+\zeta(\frac1n)=\frac12$ , The latter expressed the relationship between $\zeta(n) $ and $\zeta(\frac1n)$ with $n$ is a positive integer , Then i ...
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Imaginary component of Dirichlet Eta Function's root with real component equal to 1/2

Let $$\space\space\space \eta(z) = \sum_{a=1}^{\infty} \frac{1}{a^{z}} \cdot (-1)^{a-1} $$ Now let $\space$$z = \sigma + it $ , $$\eta(\sigma + it) = \sum_{a=1}^{\infty} \frac{1}{a^{\sigma + it}} \...
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symmetric double-integral on fractional part

Let $\{\}$ denotes the fractional part function, does the following double-integral have a closed-form ? $$\int_{0}^{1}\int_{0}^{1}\bigg\{\frac{1}{x}+\frac{1}{y}\bigg\}dx\,dy$$
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Sum over all inverse zeta nontrivial zeros

Starting from the Hadamard product for the Riemann Zeta Function (assuming the product is taken over matching pairs of zeros) $$\zeta(s)=\frac{e^{(\log(2\pi)-1-\gamma/2)s}}{2(s-1)\Gamma(1+s/2)}\prod_{\...
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Holomorphic but not a Dirichlet series

I want an example of holomorphic map $f : \Omega \to \Bbb C$ where $\Omega$ is the half-plane $\mathrm{Re}(s) > 1$, such that there is no sequence $(a_n)$ of complex numbers with $$f(s) = \sum_{n \...
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Solutions to functional equational of Riemann zeta function

Is it possible to determine all the meromorphic functions $f : \Bbb C \to \Bbb C$ which have a pole only at $s=1$, of order $1$, and such that $$f(s) = 2^s \pi^{s-1} \sin(\pi s / 2) \Gamma(1-s) f(1-...
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Is there a bounded function that is always greater than $M (t) = \max_{s \in [0, t]} \left| \zeta \left( \frac{1}{2} + i s \right) \right|$?

Is there a bounded function that is always greater than $M (t) = \max_{s \in [0, t]} \left| \zeta \left( \frac{1}{2} + i s \right) \right|$ ?
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Is there a “natural” entire series associated to the Riemann zeta function whose radius of convergence is $\frac{1}{\sqrt{5}}$?

As a follow-up to Is there a hidden connection between RH and the golden ratio?, let's consider the plane $ P $ whose intersection with the Riemann sphere is the circle I denoted by $ \Gamma_{\Delta}...
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Does $N_0(T) =N(T)$ where $N$ is the exact Riemann $\zeta$ zero counting function and $N_0$ is the approximate zero counting function imply the RH?

From A theory for the zeros of Riemann ζ and other L-functions The main new result is that the zeros on the critical line are in one-to- one correspondence with the zeros of the cosine ...
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Fractional part of the floor function Integral [closed]

Let $\lfloor\rfloor\ $ and $\{\}$ denote the floor function and the fractional part funtion, respectively. Then calculate in closed-form the following integral $$\int_{0}^{1}\bigg\{\frac{1}{x}\bigg\...
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Riemann Zeta function is Big Oh of log(|t|)

I'm stuck proving the following: for all $t$ such that $|t|$ sufficiently large and if $\sigma > 1 - \frac{100}{log |t|} $ $\zeta(\sigma + it) = O(log|t|)$ Questions: Do we only consider the ...
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The altenating series of squarefree k-almost prime zeta functions converges to the reciprocal zeta?

I have two questions. The second question is really about showing my motives to ask the first question but it would be great if someone could verify that my argument works. Let $a_k(n)$ be the $n$th ...
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Proof that $\sum\limits _{n=1}^{\infty } n\sum\limits _{j=2}^{\infty }{\frac {(-1)^{j-1}}{j^2} \left( 1-{j}^{-1} \right)^{n-1}} = -\frac12$

How can we prove the following? $-1+\frac1{12}{\pi }^{2}-\frac12\sum\limits_{n=2}^{\infty }\Gamma \left( n+1 \right) \sum\limits_{k=0}^{n+2} \,{\frac {\zeta \left( k \right) \left( - 1 \right)...
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Analytic Number Theory calculation involving zeta

In proving the weak zero free region of the zeta function given by (Hadamard, de la Vallee Poussin 1896) one has for $ \sigma' > 1$ and any $t' \in \mathbb{R}$ we have: $$|\zeta(\sigma' + it')| = ...
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References for $ \chi(n)=n\sum\limits_{j=2}^\infty\frac {(-1)^{j-1}}{j^2}\left(1-j^{-1}\right)^{n-1}$ in $\zeta$ expansion?

What is the name of these coefficients related to a series expansion for the Riemann zeta function or any other references? Did the author of this paper derive the series or get it from somewhere else?...
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How can I show that :$\sum_{n=1}^{\infty}(-1)^{n-1}(\zeta(n+1)-1))=\frac12$?

A few computations I did with Mathematica gave me this sum. I'm really very interested to know how I can evaluate $$\sum_{n=1}^{\infty}(-1)^{n-1}(\zeta(n+1)-1))=\frac12$$ I have used Direchlet eta ...
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What's this $\sum_{n=2}^{+\infty}\frac{(-1)^n}{\zeta(n)}$ equal?

This sum $\sum_{n=2}^{+\infty}\frac{(-1)^n}{\zeta(n)}$ gives from $n=2$ to odd integer negative value which is close to $-0.27,..$ and gives $0.72 ...$ to even integer, This analysis mixed me to ...
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On the integral $\int_1^\infty\big(\{x\}^n-\frac1{n+1}\big)\frac{dx}x$

According to Dirichlet's test (integral version), $$I_n=\int_1^\infty\big(\{x\}^n-\frac1{n+1}\big)\frac{dx}x$$ converges, where $n$ is a positive integer and $\{x\}$ denotes the fractional part of $x$....
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Does the zeta function have no zero with real part $> 1-\delta$?

It is known by Hadamard and de la Vallée-Poussin that there is $c>0$ such that $$\Re(s) < 1 - c/\log(2 + \Im(s))$$ for any zero $s$ of $\zeta$. My question: is it known that there is $\delta&...
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Does $z (s) = \int_0^s \zeta \left( \frac{1}{2} + i t \right) d t = s + \sum_{n = 2}^{\infty} \frac{i (n^{- i s} - 1)}{\ln (n) \sqrt{n}}$ converge?

Does $z (s) = \int_0^s \zeta \left( \frac{1}{2} + i t \right) d t = s + \sum_{n = 2}^{\infty} \frac{i (n^{- i s} - 1)}{\ln (n) \sqrt{n}}$ converge ? If we take the termwise integral for $n^{-s}$ with ...
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What are the “moments” of the Riemann zeta function?

I have been reading about the applications of the Riemann zeta function in physics and came across something called a "moment". I have never heard of such a property of the Riemann zeta function so I ...
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Is there a closed-form expression for this integral?

Does there exist a closed-form for the integral $$\int 8\frac { (\zeta(1/2+it))^2\pi}{-\Psi(1,1/4-i/2t) ( \zeta(1/2+it))^2 +\Psi(1,1/4+i/2t) (\zeta(1/2+it))^2 + 8\zeta(2,1/2+it) \zeta(1/2+it) -8(\...
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is it always possible to choose a small enough positive $\varepsilon$ such that $0 < c_n (\varepsilon) < 1$?

Let $ Y_{n, m} (t) = \left\{ \begin{array}{ll} t & m = 0\\ t + h_{n, m} \cos (\pi n) \tanh \left( \frac{Z (Y_{n, m - 1} (t))}{| \Omega (t) | \prod_{k = 1}^{n - 1} \tanh (Y_{n,...
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Infinite/Recursive Cesàro Summation of $\zeta(1)$

Is anything known about this kind of `infinite' Cesàro summation (or any related types of summation)? If we have a function we wish to sum $f(n)$, but $$ S^0[f] = \sum_{n=1}^\infty f(n) $$ diverges, ...
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How to apply the Abel-summation on this series

For a project I'm planning on using the Abel-summation to find a solution for the series I'm currently working with. These series being $$\sum_{k=1}^{\infty}((-1)^{k-1} \cdot k) = 1 - 2 + 3 - 4 + ...$$...
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Absolute maximum of Riemann Zeta Function

The question os fairly simple, though I couldn’t find its answer on the internet. Is the norm of the complex output value of the Riemann Zeta Function limited? Or can I plug in input values to make ...
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Binomial coefficient identity found in Apéry's theorem

In Apéry's proof of the irrationality of $\zeta(3)$, while proving the formula for the fast-converging series $\zeta(3)=\frac{5}{2} \sum_{k=1}^{\infty}{\frac{ (-1)^{k-1}} {k^{3}\binom {2k}{k}}}$ there ...
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Is this expression of the Riemann zeta function as a Mellin transform true for $\Im\{s\}\neq 0$?

For sure, we can write \begin{align} \zeta(s)&=\int_0^\infty \sum_{n\geq 1}\delta\big(\log \frac{x}{n}\big)x^{-s-1}dx\\ &=\int_0^\infty \sum_{n\geq 1}\delta(x-n)x^{-s}dx \nonumber \\ &=\...
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Do similar expressions exist for odd zeta values other than $3$?

Since Hjortnaes (and later Apéry), we know that $$ \zeta(3)=\frac{5}{2} \sum_{k=1}^{\infty}{\frac{ (-1)^{k-1}} {k^{3}\binom {2k}{k}}}. $$ I read somewhere that there might be a similar expression ...
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47 views

Product of Zeta function

Looking at this question on mathoverflow, I became curious about the value of $$ p=\prod_{j=2}^\infty \zeta(j)^{-1} $$ In particular, is $p>0$ (assuming it converges)?
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$ \int_0^{\infty} f(x) dx = \sqrt[5] {\zeta(3)} $ ?? [closed]

Let $f(x) $ be made of standard functions without a zeta function and with integer parameters. Then $$ \int_0^{\infty} f(x) dx = \sqrt[5] {\zeta(3)} $$ Does not have a solution. Is this ...
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Euler product on congruence classes

Is it possible to evaluate the Euler product $\prod\limits_{p\equiv 1\pmod{6}} \frac{p^3}{p^3-1}$ in terms of $\zeta(3)$? Thanks, in advance.
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A new proof that $\zeta(2n+1)$ for integer $n>1$ is irrational?

I recently came across this paper on ArXiv by N. A. Carella (an author with over 60 publications on the aforementioned website), published on 4$^{\rm th}$ June 2018, containing the following Theorem: ...
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Trying to prove that $\zeta(3)=\frac{5}{2} \sum_{k=1}^{\infty}{\frac{ (-1)^{k-1}} {\binom {2k}{k}k^{3}}}$

I am going through Van der Poorten's "A Proof that Euler Missed...", which outlines Apéry's proof that $\zeta(3)$ is irrational. In section 3. "Some Irrelevant Explanations" (page 197 in the linked ...