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Questions tagged [riemann-zeta]

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

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Integral representation of Riemann Zeta function [closed]

While messing around with integrals of infinite series I arrived with the following equation: $$\int_{0}^{\infty}\left({\frac{1}{e^{2\pi x}-1}\left(\frac{d^a }{dx^a}\left(\frac{x^{a+1}}{x^2+1} \right)\...
Matteo Gravinese's user avatar
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Is this formulation of Zeta function Valid in the Critical Strip

There are numerous equivalent formulations of Riemann Zeta function (that are valid in the critical strip). One of them I came across is: "The real and imaginary parts of the Riemann Zeta ...
stack.tarandeep's user avatar
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On how many circles will the roots of these constructed palindromic polynomials sit on?

The approximation to the Dirichlet eta function plus its conjugate: $$\left(1-\frac{1}{2^{a+i b-1}}\right) \zeta (a+b i)+\left(1-\frac{1}{2^{a-i b-1}}\right) \zeta (a-b i)=\frac{1}{1^{a-i b}}+\frac{1}{...
Mats Granvik's user avatar
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5 votes
1 answer
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Is there a reason why the Maclaurin coefficients of the Riemann zeta function are asymptotically close to -1?

If I look at the numerical values of the Maclaurin series of the Riemann zeta function I see that they approach -1 extremely quickly. In fact, if I take $\zeta(x)=\sum_{n=0}^\infty a_n x^n$ then ...
Jean Du Plessis's user avatar
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A limit involving the $n$th derivative of the reciprocal of the Riemann Zeta function: $\lim _{n \to \infty} |(1/\zeta)^{(n)}(2)|$ [closed]

Does the limit $\lim _{n \to \infty} |(1/\zeta)^{(n)}(2)|$ exist? If so, how can we determine its precise value, or is there any way to approximate it with high precision? ($(1/\zeta)^{(n)}(2)$ is ...
Haidara's user avatar
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1 answer
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Proof of zeta functional equation in Edwards

I'm trying to understand the first proof of the zeta functional equation given in Riemann's Zeta Function by H M Edwards. Referring to the excerpt from page 13 below, I'm stuck on how he derives the ...
Peter4075's user avatar
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is this solution correct $\frac {\partial}{\partial x} \int_0^∞ \frac{\sin((x+it)\arctan(t))}{((1+t^2)^{(x+it)/2} (e^{2\pi t} -1))} dt =0 $?

when I was reading about the Riemann zeta function I found out this integral $\ \frac {\partial}{\partial x} \int_0^∞ ​ \frac{\sin((x+iy)\arctan(t))}{((1+t^2)^{(x+iy)/2} (e^{2\pi t} -1))} dt $ and ...
Prateek Sharma's user avatar
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what is the value of d/ds (integral(0,infinity) sin(s arctan t)/((1-t^2)(s/2) (e^(2pi t) -1)) dt) [closed]

I was reading about the Riemann zeta function and found the integral form of it ζ(s)= 1/(s-1) + 1/2 + 2(integral(0, infinity) sin(s arctan t)/((1-t^2)(s/2) (e^(2pi t) -1)) dt). When I was trying to ...
Prateek Sharma's user avatar
-2 votes
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56 views

Finding the Taylor Series for $f\left(x\right)=\frac{x}{e^{x}-1}$ [closed]

I want to see how one would figure out that $\frac{x}{e^{x}-1}=\sum_{n=0}^{\infty}B_{n}\frac{x^{n}}{n!}$ where $B_{n}$ represents the nth Bernoulli number with the convention that $B_{1}=-\frac{1}{2}$....
TF2 Sniper Main's user avatar
7 votes
1 answer
716 views

What's the easiest way to prove the Riemann Zeta function has any zeros at all on the critical line?

I learned in my intro to complex analysis class that the Riemann Zeta has trivial zeros at the negative even integers; this follows from the Riemann Zeta Reflection identity and the behavior of the $\...
Faraz Masroor's user avatar
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1 answer
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$H(x)$ approximates $\pi(x)$ pretty well. But what are the drawbacks, when compared with Riemann's $R(x)$?

I'm aware of the Gram series which is equivalent to $R(x)$ (Riemann prime counting function): $$ R(x)=\sum_{n=1}^\infty \frac{\mu(n)}{n}li(x^{1/n}). $$ Over the interval $x=2$ to $x=10^4$ the average ...
zeta space's user avatar
-6 votes
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101 views

I have found an expression of the reciprocal of the zeta function involving prime zeta function but how can it be proven? [closed]

I have tried to find a proof of this: $$\frac{1}{\zeta (s)}=\frac{1}{1+P(s)}(1-P(s)+\sum_{n=1}^{\infty }(-1)^{n-1}\prod_{k=1}^{n}(P(s)-R_{k}(s))$$ where: $R_{n}(s)=\sum_{k=1}^{n}\frac{1}{p_{k}^{s}}$ ...
Ahmet Ali Çetin's user avatar
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1 answer
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Finding a closed form for $\sum^{\infty}_{n=1} \frac{1}{(n+1)n^\alpha}$ [duplicate]

I encountered the following sum in my work and I was wondering if it has a known closed form: $$ \sum^{\infty}_{n=1} \frac{1}{(n+1)n^{\alpha}} \quad , \quad 0 < \alpha < 1 \; , \; \alpha \in \...
Aidan R.S.'s user avatar
4 votes
2 answers
161 views

Asymptotics of $I_n:=\int_1^\infty \frac{d}{dx} [x^{3/2}\psi ' (x)]x^{-1/4}(\log x)^{2n}$ as $n\to\infty$.

In Section 1.8(3) of the monograph Riemann zeta function, author H. M. Edwards says that Riemann derived the power series $$\xi(s)=\sum_{n=0}^\infty a_{2n}\left(s-\frac{1}{2}\right)^{2n}$$ and stated ...
Alann Rosas's user avatar
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(Im)possibility of closed-form expression of Clausen functions

When I started learning Riemann zeta function, I was fascinated that $\zeta(2n)$ can be expressed with finite integers and $\pi$ while $\pi$ has no obvious relation with the sum-$\zeta(2n)$ but no &...
Quý Nhân Đặng Hoàng's user avatar
2 votes
1 answer
44 views

Evaluating the odd part of an infinite series

Let's consider the following sequence $$ a_n = \begin{cases} 0, \quad n = 2k, \, k \in \mathbb{Z} \\ \frac{1}{n^4}, \quad n = 2k+1, \, k \in \mathbb{Z} \end{cases}. $$ I would like to compute the ...
Hendrra's user avatar
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9 votes
3 answers
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"Are there any simple groups that appear as zeros of the zeta function?" by Peter Freyd; why is this consternating to mathematicians?

I would like to understand the "upsetting"-to-mathematicians nature of this question Freyd poses to demonstrate that "any language sufficiently rich that to be defined necessarily ...
Hooman J's user avatar
  • 257
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1 answer
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Integration over Hankel contour

In a write-up by Paul Garrett, he claims that he can apply the Residue Theorem to the equality $$ \zeta\left(s\right) = \frac{1}{\Gamma\left(s\right)\left(1-{\rm e}^{2\pi{\rm i}s}\right)} \lim_{\...
cxrlo's user avatar
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1 vote
0 answers
108 views

Conjectured Riemann Zeta Function Integrals, $\zeta(s)$ linked to Gregory Coefficients and convergent for $\Re(s)>0$ and $s\neq 1$

The Riemann Zeta Function $\zeta(s)$ is typically defined for complex $s$ and $\Re(s)>0$ by $$\zeta(s)=\sum_{n=1}^\infty \frac{1}{n^s}=\frac{1}{\Gamma(s)}\int_0^\infty \frac{x^{s-1}}{e^x-1}\,dx$$ ...
James Arathoon's user avatar
1 vote
0 answers
100 views

Formula for the Sum of the Reciprocals of the Imaginary Parts of the Nontrivial Zeroes of the Riemann Zeta Function

Assume the Riemann Hypothesis, so that every nontrivial zero $\rho$ of the Riemann zeta function looks like $\rho = \frac{1}{2} + \gamma i$. Let $T > 0$ be an integer. On page 51 of Appelgren and ...
Johnny Apple's user avatar
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0 votes
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Termwise Differentiation of Convergent Series Involving the Riemann Zeta Function

Let $\zeta(s)$ be the Riemann zeta function. We know that $$\zeta(s) = \prod_p(1 - p^{-s})^{-1}, \ \operatorname{Re} s > 1$$ in the sense that the righthand side converges absolutely to $\zeta(s)$ ...
Johnny Apple's user avatar
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About the $n$th derivative of the Riemann zeta function on positive even integers

I know there exist a formula for the Riemann zeta function on positive even integers involving Bernoulli numbers. Do there exist any closed form for the $n$th derivative of the Riemann zeta function ...
Haidara's user avatar
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1 vote
1 answer
63 views

Write the sum in terms of the Riemann zeta function

I believe it is a question from JHMT. Write the sum $\sum_{a=1}^\infty\sum_{b=1}^\infty\frac{gcd(a,b)}{(a+b)^3}$ in terms of Riemann zeta function. The answer should be $-Z(2)+\frac{Z(2)^2}{Z(3)}$...
user1200034's user avatar
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Stein Complex Analysis Proof of Chapter 6 Proposition 2.5

In Stein's Complex Analysis book, within the proof of Chapter 6 Proposition 2.5, the following claim is made: For $s = \sigma + it \in \mathbb{C}, n \geq 1$, then $$\left| \frac{1}{n^s} - \frac{1}{x^s}...
Mashe Burnedead's user avatar
2 votes
0 answers
66 views

Dedekind zeta function of abelian number fields is a product of Dirichlet L-functions

Can somebody give an explicit (and self-contained) proof of the fact that the Dedekind zeta function of an abelian number field $K$ is a product of Dirichlet L-functions? I.e., $$ \zeta_K(s)=\prod_{\...
Sardines's user avatar
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4 votes
0 answers
127 views

If $\zeta(5)=\frac{a}{b}\in\mathbb{Q}$ then $b\neq p^k$

If $\zeta(5)=\frac{a}{b}\in\mathbb{Q}$ then prove that $b\neq p^k$ where $p$ is any prime and $k\in\mathbb{N}$ Take $a,b\in\mathbb{N}$ such that $(a,b)=1$. Now if $b=p^k$ then $$p^k\zeta(5)=a$$ So, by ...
Max's user avatar
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1 vote
0 answers
95 views

Integral involving the zeta function along the critical line

WolframAlpha claims that $$\int\limits_{-\infty}^{\infty}\frac{|\zeta(\frac{1}{2}+it)|(3-\sqrt{8}\cos(\ln(2)t))}{t^2 +\frac{1}{4}}dt=\pi\ln(2)$$ and I don't know how to show this. I tried removing the ...
Darmani V's user avatar
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0 answers
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Inequalities for the Dirichlet eta function at non-trivial Riemann zeta zeros.

I am interested in these inequalities for sufficiently large $n$: $$\Large \left(\Re\left(\sum _{k=1}^n (-1)^{k+1} x^{\log (k) c}\right)\right)^2 \leq \left( \Re\left(x^{\log \left(n+\frac{1}{2}\right)...
Mats Granvik's user avatar
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4 votes
1 answer
140 views

Closed form of $\int_{(0,1)^5} \frac{xyuvw}{(1-(1-xy)u)(1-(1-xy)v)(1-(1-xy)w))}\, dx\,dy\,du\,dv\,dw$

I need closed form for the integral $$I:=\int_{(0,1)^5} \frac{xyuvw}{(1-(1-xy)u)(1-(1-xy)v)(1-(1-xy)w))} \, dx\,dy\,du\,dv\,dw$$ Then $$0<I<\int_{(0,1)^5} \frac{1}{(1-(1-xy)u)(1-(1-xy)v)(1-(1-xy)...
Max's user avatar
  • 910
3 votes
0 answers
88 views

Infinite summation of a falling factorial divided by a power

I'm trying to find the result of this summation $$ S(r,m)=\sum_{n=m}^\infty \frac{(n)_m}{n^r} $$ where $(r,m\in\mathbb{N})$, $(r\geq3)$, $(1\leq m \leq r-2)$ and $(n)_m=\frac{n!}{(n-m)!}$ is a falling ...
Max Pierini's user avatar
2 votes
0 answers
79 views

Zeta Function Zeros and Dedekind Eta Function Integral

Working on algorithms related to number theoretical function calculation performance improvement and accidentally discovered the following for Dedekind $\eta$ function: $$-\int_1^{\infty } \left(t^{-s-...
Gevorg Hmayakyan's user avatar
1 vote
1 answer
582 views

Breakthrough on zero density estimates on Riemann hypothesis

Terence Tao announced a breakthrough on Riemann hypothesis Original paper by Guth and Maynard. Tao writes: Let $N(\sigma,T)$ denote the number of zeros of the Riemann zeta function with real part at ...
zeynel's user avatar
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0 votes
1 answer
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How does the Cauchy Condensation Test prove that for any $p > 1$, the sum $\sum \frac{1}{n^p}$ converges [closed]

In a previous answer on MSE, it was stated that using the Cauchy Condensation Test and the convergence of a geometric series, it can be shown that if $p > 1$, the follow sum converges: $$\sum_{n=1}^...
Larry Freeman's user avatar
1 vote
0 answers
68 views

Divergence of the Riemann Zeta function

Consider the equation $\displaystyle\zeta\left(s\right) \Gamma\left(s\right) = \int_{0}^{\infty}\frac{u^{s - 1}}{{\rm e}^{u} - 1} {\rm d}u$. This integral equals $\displaystyle\int_{0}^{\infty} \frac{...
Aspirant29's user avatar
1 vote
1 answer
82 views

May I find $\zeta(-1)$ using the Hankel formula for $\zeta$, but not the reflection formula?

The Reimann zeta function for $\Re(s) > 1$ is $$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s},$$ and we may show that the analytic continuation of $\zeta$ onto $\Re(s) \leq 1, s \neq 1$ is $$\zeta(s)...
Robin's user avatar
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2 votes
2 answers
275 views

On the connection between zeros of the Riemann Zeta function and prime numbers - a simple graphical approach

Premise What follows is a simple approach to understand something more about linking prime numbers and continuous functions / summations by someone who does not know complex analysis but is fascinated ...
Matteo's user avatar
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0 answers
36 views

Is there a function that gives an estimate of how many zeros of riemanns zeta function occur before a given prime number?

The prime counting function tells us how many primes occur before a given number e.g x/ log x but is there a similar approximation for how many zeros there are before a given orime or even before any ...
user avatar
2 votes
0 answers
43 views

Comparing two series expressions for $1/\zeta(s)$. What can be said about their complex roots?

The following two expressions involving the inverted Riemann $\zeta(s)$ functions are well known: \begin{align} \frac{1}{\zeta(s)} &= \sum_{n=1}^\infty \frac{\mu(n)}{n^s} \\ -\frac{\zeta'(s)}{\...
Agno's user avatar
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4 votes
1 answer
56 views

What does "behaving like independent random variables" mean?

The following is from Gonek's paper: The situation is much more interesting if the Riemann hypothesis holds. In that case we may rewrite (6) as $$ \sum_{0 < γ \le T} x^{iγ} « Tx^{-1/2+ Ɛ} + x^{1/2+...
Ali's user avatar
  • 281
4 votes
1 answer
119 views

$3\frac{\zeta'(\sigma)}{\zeta(\sigma)}+4 \Re \frac{\zeta'(\sigma+i t)}{\zeta(\sigma+i t)}+\Re\frac{\zeta'(\sigma+2 i t)}{\zeta(\sigma+2it)}\leq0$

In our script it is used without proof that For $\sigma>1$ and $t \in \mathbb{R}$ $$ 3 \frac{\zeta^{\prime}(\sigma)}{\zeta(\sigma)}+4 \Re \frac{\zeta^{\prime}(\sigma+i t)}{\zeta(\sigma+i t)}+\Re\...
calculatormathematical's user avatar
1 vote
2 answers
57 views

Summation form of improper integrals

On page 9, Edwards has this expression $$ \int_0^{\infty} e^{-nx} x^{s-1} dx = \frac{\Pi(s-1)}{n^s}$$ obtained from Euler’s factorial formula by replacing $x$ with $nx$. Can you help with the next ...
zeynel's user avatar
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0 votes
1 answer
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Estimation of the absolute value of the $n$th non-real zero of the Riemann zeta function

Recently, I have been studying the oringinal proof of the prime number theory by Hadamard. I didn't get it on the estimation of the absolute value of the $n$th non-real zero of the $\zeta$ function by ...
Derek Xie's user avatar
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0 answers
72 views

the zero's of $f(s,a) = \sum_{n=1}^{a-1} n^{-s} $

I was looking at the zero's of $$f(s,a) = \sum_{n=1}^{a-1} n^{-s} $$ for integer $a>3$ in the strip $0 < \operatorname{Re}(s) < 1$. Now this clearly relates to the Riemann zeta: $$f(s,a) + \...
mick's user avatar
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0 votes
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Euler's gamma function and Riemann zeta

On page 7 Edwards gives Euler’s factorial function: $\displaystyle n! = \int_0^{\infty} e^{-x} x^n dx$ for $n=1,2,3,\ldots $ On the next page he gives the same in Gauss’ notation $\displaystyle \Pi(s) ...
zeynel's user avatar
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0 votes
0 answers
37 views

Show $\frac{1}{\pi^n}\zeta(n) \in \mathbb{Q}$ for even n

I am working on an exercise with aim to prove $\frac{1}{\pi^n}\zeta(n) \in \mathbb{Q}$ using Fourier series. Here's the outline: We first let $f_1(x) = x-\frac{1}{2}$, then we can express it with ...
aawangas's user avatar
1 vote
0 answers
34 views

maximal continuation of $\Pi_2(x)$

Consider the functions for $k\in \Bbb N$ $$ \Pi_k(x) := \sum_{n \in \Bbb N} e^{\frac{\log^k n}{\log x}} $$ $\Pi_1(x)$ converges for real $1/e<x<1$. $\Pi_1(x)$ is a Riemann zeta function i.e. $\...
zeta space's user avatar
0 votes
1 answer
61 views

Lower bound for the prime zeta function

The prime zeta function is defined for $\mathfrak{R}(s)>1$ as $P(s)=\sum_{p} \ p^{-s}$, where $p \in \mathbb{P}$. It is well-know this series converges whenever $\mathfrak{R}(s)>1$. Now, ...
Frank Vega's user avatar
1 vote
1 answer
62 views

Proving that $\left|\sum_{n<x}\mu(n)\right|\ll x\exp(-c\sqrt{\log x})$ for some $c>0$

Assume that for $\sigma\ge 1-\frac{1}{(\log(2+|t|)^2}$ we have $$|\zeta(\sigma+it)|\gg\frac{1}{(\log(2+|t|))^2}.$$ Using Perron's formula and moving the line of integration to $\textrm{Re}(s)=1-\frac{...
turkey131's user avatar
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0 votes
0 answers
56 views

How do we test Riemann zeta zeros for simplicity?

I understand that we call a “simple zero” if the first derivative of the complex function $\neq 0$. How does this apply to the zeros of the Riemann zeta? I read that all known zeros are simple. Taking ...
zeynel's user avatar
  • 447
0 votes
1 answer
36 views

Proving $|\zeta(\sigma+it)|=O(\log t)$ uniformly for $1\le\sigma\le 2$ and $t\ge 10$

Prove that $|\zeta(\sigma+it)|=O(\log t)$ uniformly for $1\le\sigma\le 2$ and $t\ge 10$. The solution given by my lecturer is as follows. Recall the approximate formula for zeta, given by $$\zeta(s)=\...
turkey131's user avatar
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