Questions tagged [riemann-zeta]

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

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Dirichlet Series of a given sequence

I am trying to calculate the dirichlet generating function of $(p(n)q( \log(n)))_{n \geq 1}$ where $p,q$ are arbitary polynoms. First I calculated the dirichlet genarating function of $(p(n))_{n \geq ...
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New formulae for the Riemann Zeta Series

Back in late April I arrived at the following formula for when $x+b < -1$: $$\boxed{ \sum_{a=1}^{\infty}\sum_{k=0}^{\infty}\frac{(k+a)^{x+b}}{a} = \sum_{k=0}^{\infty}\sum_{g=0}^{k}\frac{b^{g}}{(g!)...
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Riemann Hypothesis and Electric Dipole

I recently watched 3Blue1Brown's video on the Riemann zeta function (excellent video by the way), and basically he shows what analytic continuation is and how it relates to the function. In order to ...
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2 votes
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How do I obtain this expression for the Taylor series of$\frac{z^2}{\sin ^2(z)}$?

Reading about Complex Analysis, I came across the following: Consider first the representation $\frac{\pi ^2}{\sin ^2(\pi z)}=\sum_{n\in \mathbb{Z}}\frac{1} {(z+n)^2}$, which applies for all $z\in \...
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Is $f(x)$ worse than $\text{Li(x)}$ at counting primes?

This is the Gram series: $$G(x)=1+\sum_{k=1}^\infty\frac{(\ln x)^k}{kk!\zeta(k+1)}$$ It is equivalent to the Riemann prime counting function: $$R(x)=\sum_{n=1}^\infty \frac{\mu(n)}{n}li(x^{1/n})$$ I ...
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A double series nature

here is defined a double series: $\displaystyle \sum\limits_{i=1}^{n}\sum\limits_{j=1}^{m} \frac{\cos(b\ln(2i))\cos(b\ln(2j))}{(2i)^{a}(2j)^{a}}-\frac{\cos(b\ln(2i-1))\cos(b\ln(2j-1))}{(2i-1)^{a}(2j-1)...
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Are the non trivial zeros of Riemann's Zeta function rational, irrational or we don't know? [duplicate]

I've read and watched from several sources that all non trivial zeros are hypothesized to be on the critical line (vertical line at x=1/2), and that, so far, nobody has been able to find one outside ...
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A curious limit: $\lim\limits_{n\to\infty}\sum\limits_{i=1}^{n}\left[\left(\frac{n}{n+1-i}\right)\right]^{a}f(i) = c\sum\limits_{i\geq 1}f(i)$

I am trying to prove, for the general case whereby $\zeta(\cdot\,,\cdot)$ is the Hurwitz-Zeta function, and $a\in \mathbb{N}$, that $$\mathcal{L} = \lim\limits_{n\to\infty^{+}}\sum\limits_{i=1}^{n}\...
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Is there a closed-form expression for $f(x)=\sum\limits_{k=1}^\infty (\text{Ei}(i k x)+\text{Ei}(-i k x))$ assuming $x\in\mathbb{R}$?

Question: Is there a closed-form expression for $$f(x)=\sum\limits_{k=1}^\infty (\text{Ei}(i k x)+\text{Ei}(-i k x))\,,\quad x\in\mathbb{R}\tag{1}$$ where $\text{Ei}(z)$ is the exponential integral ...
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Recursive analytic continuation of Riemann zeta function

If you read pages 51-55 of the book The Theory Of Functions by Konrad Knopp (Publication date 1947) and you are patient enough to overcome a very bad made digital copy of the book (I do not understand ...
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Question on term of derived formula for $\log\zeta(s)$

The derived formula $$\log\zeta(s)=-\underset{N\to\infty}{\text{lim}}\left(\sum\limits_{n=1}^N 1_{n\in\mathbb{P}} \left(2 \tanh ^{-1}\left(1-2 n^s\right)-i \pi\right)\right),\quad s>1\tag{1}$$ ...
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Speed of Convergence for some series (double sum)

For $\alpha>2$, i want to find the speed of convergence of $$S_n=\sum_{i=1}^n\sum_{j=n-i+1}^n i^{-\alpha}j^{-\alpha}.$$ In particular, i want $\sum_{i=1}^n\sum_{j=n-i+1}^n i^{-\alpha}j^{-\alpha}\...
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Formulas of Mellin inversion theorem that involve Riemann zeta function $\zeta (s)$ and floor function $\lfloor x\rfloor$

Functions $f(x)=\lfloor x\rfloor$ and $g(s)=\frac{\zeta (s)}{s}$ are related by Mellin inversion theorem, for $c>1$, $\Re(s)>1$. $$\mathcal{M}_x(f(x))(s)=\mathcal{M}_s^{-1}(g(s))(x)$$ $$\tag{1.1}...
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Questions on summatory function related to non-integer-powers

Consider the summatory function $$f(x)=\sum\limits_{n=1}^x 1_{n\ne k^m}\tag{1}$$ where $1_{n\ne k^m}$ is the non-integer-power indicator function which returns $1$ when $n$ is a non-integer-power and $...
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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 ...
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Integral representation for $\zeta(3)$

It is well-known that $$10\int_0^{\ln \phi} t^2\coth t dt =\frac{5}{2}\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^3\binom{2n}{n}}=\zeta(3),$$ where $\phi=(1+\sqrt{5})/2$ (see Alfred van der Poorten, "A ...
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Convergence issues with this derivation of the functional equation of the Riemann zeta function

I am using Riemann's Zeta Function by H.M. Edwards as a reference for the derivation of the functional equation of the Riemann Zeta Function, as well as this pdf. I am not very familiar with the ...
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Question about regularised product of primes:

We know "super-regularized"( the term coined by authors of the paper: http://cds.cern.ch/record/630829/files/sis-2003-264.pdf) product of primes $4π²$ i.e. $$\infty \# = \prod_{k=1}^\infty ...
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Asymptotic formula for $\sum_{n<x} \frac {d(n)}{\sqrt{n}}$

I am looking for symptotic formula for $\sum_{n<x} \frac {d(n)}{\sqrt{n}}$ which doesn't use $\zeta(\frac{1}{2})$ My guess is - perhaps it is something like $A\sqrt{x}\log{x} + B \sqrt{x} + O(\log{...
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Efficient computation of $\prod_{p\equiv a\pmod m}(1-p^{-s})^{-1}$

Let $0<a<m$ be integers with $\gcd(a,m)=1$. For $s\in\mathbb{C}$ with $\Re s>1$, define $$\zeta(m,a;s)=\prod_{\substack{p\text{ is prime}\\p\equiv a\pmod m}}(1-p^{-s})^{-1}.$$ I'm looking for ...
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$\lim_{s \to 1^+} 1/\zeta(s) = 0$ obvious or not?

I read the statement that $$\frac{1}{\zeta(s)} = \sum_{n=1}^\infty \frac{\mu(n)}{n^s} \textrm{ for } \Re(s) > 1 \qquad (*)$$ In fact I can guess what the proof is: just expand both $\zeta$ and the ...
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How to prove $\int_0^\infty\frac {\tanh(x)-x\exp(-x)}{x^2}dx=\frac{\zeta'(0)}{\zeta(0)}-\frac{\zeta'(2)}{\zeta(2)}+\gamma-\frac73\log(2)$?

By educated guessing, inspired by this solution of $\int_0^\infty\frac {\tanh^3(x)}{x^2}dx$, I have found numerically: $$\int\limits_0^\infty\frac {\tanh(x)-x\exp(-x)}{x^2}dx=\frac{\zeta'(0)}{\zeta(0)}...
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Different ways to average an arithmetic function

Consider the following ways to average a multiplicative arithmetic function $f$ over $\mathbb{N}$: Arithmetic: $\displaystyle\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N f(n)$ Factorized: $\displaystyle\...
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Prove that Riemann zeta function $\zeta(z)$ is defined for $z \in \mathbb{C}$ such that $Rez > 1$

Prove that Riemann zeta function $\zeta(z)$ is defined for $z \in \mathbb{C}$ such that $Re(z) > 1$. $$\zeta(z) = \sum_{n=1}^\infty \frac{1}{n^z}$$ For real values bigger than one it's obvious ...
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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 ...
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Is there any analysis for this series $\phi=\sum_{n_1, n_2,n_3 \in Z}^\infty e^{-\sqrt{n_1^2 + n_2 ^2+n_3^2}}$

Is there any analysis for this series $\phi=\sum_{n_1, n_2,n_3 \in Z}^\infty e^{-\sqrt{n_1^2 + n_2 ^2+n_3^2}}$? This one seems like an extension of Riemann Theta Function where it takes a square root ...
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Function used to represent the Riemann Zeta function is analytic

Consider the function $g(z)= \sum^{\infty}_{n=1}\int^{n+1}_{n}\frac{t-n}{t^{z+1}}dt$. I want to show that this defines an analytic function for Re(z)>0. Later I will use it to describe the Riemann ...
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calculate the integral $\int_{0}^{+\infty} \frac{t^{z-1}\cos(t)}{e^{t}-1} dt$

I am trying to calculate the integral $$\int_{0}^{+\infty} \frac{t^{z-1}\cos(t)}{e^{t}-1} dt$$
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3 votes
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A ‘quite’ elementary function $f(x)=\frac{x^{s-1}}{s^{x}-1}$ related to Riemann-Zeta function

So I was studying the following function: $f(x)=\frac{x^{s-1}}{s^{x}-1}$ where $s$ is any natural number greater than $1$ by playing with Wolfram-Alpha a bit I observed that by taking the integral of $...
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How fast does $\prod_{p\leq x}\frac 1{1-p^{-k}}$ diverge when $0<k<1$?

We have that $$\limsup_{n\to\infty}\frac{\sigma(n)}{n\log(\log(n))}=e^\gamma$$ $$\limsup_{n\to\infty}\log(\tau(n))\frac{\log(\log(n))}{\log(n)}=\log(2).$$ I'm trying to find a similar result for $\...
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Question on convergence of explicit fomulas for summatory functions related to Dirichlet series

Given the totient summatory function $$\Phi(x)=\sum\limits_{n=1}^x\varphi(n)\tag{1}$$ and the related Dirichlet series $$\frac{\zeta(s-1)}{\zeta(s)}=\underset{N\to\infty}{\text{lim}}\left(\sum\limits_{...
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2 votes
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Irrationality Result

I have recently derived the following result: for any odd positive integer $k$, one of $\zeta(k)$ and $\zeta(k,2/3)$ is irrational. I cannot find this result anywhere. Is it already known? Thank you ...
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Formula connecting $\zeta(s)$ with $\Gamma(s,iz)$

Does some know where it came from this formula $a\sum_{k=1}^{\infty}\left(\frac{{\mathrm{e}}^{2\pi\mathrm{i}k(z+h)}}{\left(2% \pi\mathrm{i}k\right)^{a+1}}\Gamma\left(a,2\pi\mathrm{i}kz\right)+\frac{{% ...
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Dirichlet Series of Square Full Integers.

As in the title, I want to find the Dirichlet series $F$ of the indicator function for cube full integers $f(n)=1 \iff p^3|n, \forall p|n$ and $f(n)=0$ otherwise. Since $f$ is clearly multiplicative, $...
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Proving that $|\zeta(\sigma+it)|^{-1}\leqslant 4$

I am asked to prove that $|\zeta(\sigma+it)|^{-1}\leqslant 4$ for all $\sigma\geqslant 2$ and $|t|\geqslant 1$. Of course, one can use the fact that $$ \zeta(\sigma+it)^{-1}=\sum_{n=1}^{+\infty}\frac{\...
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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.
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References for Littlewood's "infinitely many crossovers" theorem from 1914

I was looking into Littlewood's 1914 result that pi(x) and Li(x) cross infinitely many times, and I came across this Wikipedia page: https://en.wikipedia.org/wiki/Skewes%27s_number#Riemann's_formula. ...
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2 votes
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A question on the Zeta function of Riemann

I am reading a paper "A note on the Riemann zeta function" by F. T. Wang, Bull. Amer. Math. Soc $(1946)$. Let $K$ be the unit semi circle with center $z=0$ lying in the right half plane $\Re(...
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Magnitude of Riemann zeta function on horizontal strip.

Background information: I am doing a problem from my analytic number theory homework, which is asking me to prove the following: Suppose $s=\sigma +it$,$0<\sigma\leq 1$, $|t|\geq 2$. Suppose $z=x+...
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Assuming Riemann's Hypothesis, is the function $\log\zeta(s)$ continuous on the critical strip except for Zeta zeros on the critical line? [closed]

Assuming Riemann's Hypothesis, is there a branch of the logarithm for which the function $\log\zeta(s)$ is continuous on the critical strip except for Zeta zeros on the critical line ? Assuming ...
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Multiplicities of zeros of the zeta function

It is well known that if $\rho$ is a nontrivial zero of $\zeta(s)$ then so are $1-\rho$, $\bar{\rho}$, and $1-\bar{\rho}$. My question is: do these four zeros necessarily have the same multiplicity? I ...
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2 votes
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Where does Laurent series for zeta function converge?

I am wondering in what region of the complex plane the Laurent series $$ \zeta(s)=\frac{1}{s-1} + \sum_{k=0}^{\infty} \frac{(-1)^k \gamma_k}{k!} (s-1)^k $$ converges. It is straight forward to derive ...
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1 answer
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How large can the jumps of $ \frac1\pi\arg\zeta\left(\frac12+iT\right) $ be?

It is known that the function $ \frac1\pi\arg\zeta\left(\frac12+iT\right) $ is continuous, except when T is the imaginary part of a Zeta zero. In that case the jump of this function can only be ...
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Integral in Edwards' book

I would like to understand the proof that $$\lim_{\tau\rightarrow \infty} \lim_{T\rightarrow \infty} \int_{a-iT}^{a+iT}\frac{\log(1-s/(\sigma+i\tau))}{s^2}x^s ds =0$$ where $a$ and $\sigma$ are real ...
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$\oint \ \frac{ \zeta'(z)} {\zeta(z)}dz$ should give number of zeroes of $ \zeta(s)$ times $2\pi i$ but, due to some reasons, it isn't.

Basically, if we integrate $\oint \ \frac{ \zeta'(z)} {\zeta(z)}dz$ over a rectangle of vertices, say (1+15i), (0+15i), (0+10i) and (1+10i) (counter-clockwise), then, according to Cauchy's Residue ...
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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{...
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Evaluate: ${{\int_{0}^{1}\frac{\ln(1+x)^5}{x+2}dx-\int_{0}^{1}\frac{\ln(1+x)^5}{x+3}dx+5\ln2\int_{0}^{1}\frac{\ln(1+x)^4}{x+3}dx}}$

Evaluate: $${{I=\int_{0}^{1}\frac{\ln(1+x)^5}{x+2}dx-\int_{0}^{1}\frac{\ln(1+x)^5}{x+3}dx+5\ln2\int_{0}^{1}\frac{\ln(1+x)^4}{x+3}dx.}}$$ The answer is given below: $$ I=-\frac{7}{12}\pi^4\ln^2(2)-\...
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1 vote
1 answer
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Can this formula for $\zeta(3)$ be proven or simplified further?

This question is related to the equivalence of formulas (1) and (2) below where formula (1) is from a post on the Harmonic Series Facebook group and formula (2) is based on evaluation of the integral ...
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How to guarantee uniqueness of analytic continuation of the zeta function

The Riemann zeta function has one analytic continuation to $\Re(s)>0$ given by: $$\zeta(s)=\frac{s}{s-1}-s\int_{1}^{\infty}\frac{\{x\}}{x^{s+1}}dx \space \space (1)$$ It also has anther analytic ...
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-1 votes
1 answer
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confused by tutorial/example on uniform convergence of $\zeta(x)=\sum 1/n^x$

I am confused by a tutorial/example on uniform convergence of $\zeta(x)=\sum 1/n^x$. The source is a reputable and widely used website: https://brilliant.org/wiki/uniform-convergence/. An screenshot ...
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