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

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3
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2answers
30 views

What is already known for $\zeta(n)$, $n\in 2\mathbb{N}+1$

Apéry showed $\zeta(3)\notin\mathbb{Q}$. What is also known or conjectured for $\zeta(n)$ with n odd? Is for example something known for $\zeta(5)$? Is there a theorem that says 'at least one of $\...
2
votes
0answers
46 views

Why are we interested in such things as $\zeta(3)\notin\mathbb{Q}$? [on hold]

In 17xx Euler gived a formula for the real numbers $\zeta(2n),~n\ge 1,$ which showed the irrationality of $\zeta(2n)$. In 1975 Apéry showed $\zeta(3)\notin\mathbb{Q}$. Why are we interested in such ...
2
votes
1answer
65 views

Question on a proof of $\zeta(3)\notin\mathbb{Q}$

I have a question on this article proving $\zeta(3)\notin\mathbb{Q}.$ by using modular forms. This is theorem 1 at page 275 (page 5 in the pdf). Most things in the proof are clear but I don't get the ...
0
votes
1answer
49 views

When did Euler find his formula for $\zeta(2n)$

Does anybody know when Euler found his famous formula $$\zeta(2n)=\frac{(-1)^{n-1}(2\pi)^{2n}B_{2n}}{2(2n)!}?$$
1
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0answers
48 views

Xi Function on Critical Strip - Mellin Transform

Story I'm trying to prove the following identity $$\int_0^\infty \frac{\Xi(t)}{t^2 + \frac{1}{4}} \cos(xt) dt = \frac{1}{2} \pi (e^{\frac{1}{2}x} - 2e^{-\frac{1}{2}x} \psi(e^{-2x}))$$ where $$\psi(...
0
votes
1answer
55 views

Contour Integration, Riemann Zeta (-n)

I was reading Riemann's Zeta Function by H. Edwards, and could not understand the equation on the page 12. \begin{align*} \zeta(-n) &= \frac{\prod(n)}{2\pi i}\int_{+\infty}^{+\infty} \frac{(-x)^{-...
2
votes
0answers
34 views

Convergence of sequence with $\zeta$ function

Last time I heard interesting question. Unfortunately I do not have idea how to solve it, so I decided to give it here. Let us define sequence $a_n=(\underbrace{\zeta\circ...\circ \zeta}_{n})(\pi)$ ...
0
votes
2answers
86 views

Riemann zeta function functional equation proof explanation

In Riemann zeta function functional equation proof I arrived to a following equation $$\frac{\Gamma\left(\frac{s}2\right) \zeta(s)}{\pi^{\frac{s}2}}=\sum_{n=1}^\infty \int_0^\infty x^{\frac{s}2-1}e^{-...
1
vote
0answers
55 views

Is this Dirichlet series generating function of the von Mangoldt function matrix correct?

Let $\mu(n)$ be the Möbius function and let $a(n)$ be the Dirichlet inverse of the Euler totient function: $$a(n) = \sum\limits_{d|n} d \cdot \mu(d)$$ Let the matrix $T$ be defined as: $$T(n,k)=a(...
1
vote
1answer
48 views

Searching Riemann zeta zeros in nuclear data files.

Is this match between some of the truncated Riemann zeta zeros and numbers in nuclear calculation data only a coincidence? I calculated these numbers from the Riemann zeta zeros and looked them up in ...
2
votes
1answer
89 views

Ramanujan's divergent series

I tried to prove this sum by myself, but I couldn't. $1 + 4 + 9 + 16 + ... = 0$ First, I know this sums are a bit problematic, as we can't just $'='$ an infinite sum, but I would like to see the ...
0
votes
0answers
49 views

What's about the derivative of the Riemann zeta function?

The derivative of the Riemann Zeta function is $$\zeta'(s)=-\sum_{n=2}^\infty\frac{\log n}{n^s}$$ for $\Re s>1$. Question. Can you refers us in a short post, from a divulgative viewpoint (but ...
3
votes
0answers
100 views

counterexample to RH; how big would it have to be?

If the Riemann hypothesis is false, then there has to be a first counterexample for $\zeta(z)=0$ in the critical strip with $\Re(z) \ne \frac{1}{2}$. For such a counterexample, how large would $T=|\...
5
votes
3answers
99 views

Prove $\int_0^{\infty} \frac{x^2}{\cosh^2 (x^2)} dx=\frac{\sqrt{2}-2}{4} \sqrt{\pi}~ \zeta \left( \frac{1}{2} \right)$

Wolfram Alpha evaluates this integral numerically as $$\int_0^{\infty} \frac{x^2}{\cosh^2 (x^2)} dx=0.379064 \dots$$ Its value is apparently $$\frac{\sqrt{2}-2}{4} \sqrt{\pi}~ \zeta \left( \frac{1}...
0
votes
1answer
36 views

Why can't an argument for the Riemann Zeta function be 1? What happens if we take Re(s)=1? [duplicate]

If $s=1$, then the series equals to $\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+...\to \infty$ This certainly does seem to be a convergent series. Why doesn't it have a limit?
2
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0answers
25 views

Analyitc Continuation of Partial $\zeta$ function

Let $A\subset \mathbb{N}$. Consider the partial $\zeta$ function $$\sum_{a\in A} \frac{1}{a^s}$$ We know this converges for Re$(s)>1$. Under what conditions of $A$, can this be analyically ...
0
votes
0answers
29 views

Zeta function Re(s)=1 [duplicate]

Here I am considering the original zeta function (not the extended one)$$\zeta(s)=\sum_{n=1}^\infty \frac{1}{n^s}$$ When Re(s)>1, the Zeta function converges, and if Re(s)<1 it diverges. Here is ...
2
votes
1answer
75 views

On a second-order differential inequality involving the Dirichlet eta function

After that I've tried understand the problem 6416 [1983, 60] A Second-Order Differential Inequality proposed by Sandford S. Miller in the American Mathematical Monthly (myself proposal is ...
0
votes
0answers
23 views

How to prove a self-recirpocal polynomials $P(z)$ to have all its zeros on the unit circle $|z|=1$?

Let $m(n)=10(n+1)^3$ and $$c_j(n)=\frac{2 (2j+1)}{\Gamma(j)}\sum_{k=1}^{n}(\pi k^2)^{j}\tag{1}$$, $$P(z)=\sum_{j=1}^{m(n)}(-1)^jc_j(n)\left(z^{4j+1}+z^{-(4j+1)}\right)\tag{2}$$ $$Q(z)=z^{4m(n)+1}P(z)=...
1
vote
0answers
61 views

Proving this formula for the Zeta function?

Could some one link me to a proof of this integral? $\zeta{(s)} = \frac{1}{\Gamma{(s)}}\int_{0}^{\infty} \frac{x^{s-1}}{e^x - 1} dx$ All the sites I've seen so far just introduce with the definition ...
0
votes
0answers
18 views

convergence of $\sum_{n=1}^{\infty}\frac{\sigma_k(n)}{n^s}=\zeta(s)\zeta(s-k)$

The identity $$\sum_{n=1}^{\infty}\frac{\sigma_k(n)}{n^s}=\zeta(s)\zeta(s-k)\qquad (1)$$ is well-known and valid for $s\in\mathbb{R}$ with $s>\max\{1,1+k\}$. $\sigma_k$ is the divisor function. ...
1
vote
0answers
41 views

Mobius function related series and Dirichlet summation

A well known identities of rare beauty is that: $$\sum_{n=1}^\infty \frac{\mu(n)}{n^s}=\frac 1{\zeta(s)}$$ Where $\mu(n)$ is the Mobius function and $\zeta(s)$ is the Riemann Zeta function. So, in ...
0
votes
1answer
31 views

On $\zeta(s)=h^2(s)$ when we presume that the Riemann zeta function has no zeros for $\Re s>\frac{1}{2}$

By specialization of a theorem from complex analysis, one has that on assumption that the Riemann zeta function $\zeta(s)$ has no zeros with $\Re s>\frac{1}{2}$, then there exists an analytic ...
2
votes
0answers
58 views

$\zeta(1)=\frac12\ln(2)$? Did I do something wrong?

I attempted to calculate $\zeta(1)$ and I got $\frac12\ln(2)$. $$\zeta(1)=\lim_{\epsilon\to0}\frac{\zeta(1+\epsilon)+\zeta(1-\epsilon)}2$$ $$=\lim_{\epsilon\to0}\frac{\frac1{1-2^{-\epsilon}}\eta(1+\...
1
vote
2answers
91 views

Finding a closed form for $\sum_{k=1}^{\infty}(-1)^{n-1}\frac{\zeta(2n+1)}{2^{2n+1}}$

I've been playing with series involving odd values of the zeta function. Some time ago I found the following closed form $$ \sum_{k=1}^{\infty}\frac{\eta(2k+1)}{2^{2k+1}}=\frac{1}{2}-\ln(2) $$ and ...
0
votes
1answer
42 views

Particular values of the Riemann zeta function.

On the wikipedia, near the bottom of the "Specific Values" section, there is a statement that bothers me. $$\zeta(-13)=\zeta(-1)$$ Firstly, it is well noted that the summations must be evaluated ...
1
vote
1answer
58 views

Evaluate the indefinite integral $\int \frac{t\sin at}{b^2+t^2}dt$

It is known DLMF (25.2.8) that for $\Re s>0$ and for integers $N\geq 1$ $$\zeta(s)=\sum_{k=1}^N\frac{1}{k^s}+\frac{N^{1-s}}{s-1}-s\int_{N}^\infty \frac{x-\lfloor x \rfloor}{x^{s+1}} dx,$$ where $\...
15
votes
1answer
231 views

Connection between the area of a n-sphere and the Riemann zeta function?

The Riemann Xi-Function is defined as $$ \xi(s) = \tfrac{1}{2} s(s-1) \pi^{-s/2} \Gamma\left(\tfrac{1}{2} s\right) \zeta(s) $$ and it satisfies the reflection formula $$ \xi(s) = \xi(1-s). $$ But the ...
4
votes
0answers
84 views

Is right this application of Hadamard three-lines theorem for $ \frac{\zeta(s)}{s}- \frac{d\zeta(s)}{d\sigma}$?

Let the complex variable $s=\sigma+it$, then from the following identity valid for $\sigma=\Re s>1$ $$\zeta(s)=s\int_1^\infty \frac{[x]}{x^{s+1}}dx$$ where $\zeta(s)$ is the Riemann Zeta function, ...
2
votes
1answer
90 views

How to find the bound of this sum?

Let $t>0,a(t)=\arg(\Gamma(1/4+it))$,$\kappa(n)=\frac{1}{2}x\pi n^2$,we need to calculate the bound,$A(x)$, of the following finite sum: $$ S(x)=\sum_{1\le n\le x}e^{\kappa(n)}\left(e^{ia(t)}(\kappa(...
6
votes
4answers
211 views

Limit of $\sqrt{\frac{\pi}{1-x}}-\sum\limits_{k=1}^\infty\frac{x^k}{\sqrt{k}}$ when $x\to 1^-$?

I am trying to understand if $$\sqrt{\frac{2\pi}{1-x}}-\sum\limits_{k=1}^\infty\frac{x^k}{\sqrt{k}}$$ is convergent for $x\to 1^-$. Any help? Update: Given the insightful comments below, it is ...
1
vote
0answers
24 views

How is justified the derivation under the integral sign $\frac{d}{d\sigma} \left( \Re\frac{1}{\zeta(s)} \right) $?

Taking $\sigma=\Re s>1$ (this is we take $s=\sigma+it$, $\sigma$ and $t$ real numbers) then the using theknown integral representation for $\frac{1}{\zeta(s)}$, where $\zeta(s)$ is the Riemann ...
0
votes
2answers
43 views

$\zeta(0)=-\frac{1}{2}$

How can I proove $\zeta(0)=-\dfrac{1}{2}$ only with the fact $\zeta(s)=\sum_{n=1}^{\infty}n^{-s}$ for $\mathbb{R}\ni s>1$? Does Euler's formula for $\zeta(2n),~\mathbb{N}\ni n>0$ holds also for $...
0
votes
0answers
22 views

representation of $\zeta(s)$ with line integral

Let $0<\varepsilon<2\pi$ and $\gamma_{\varepsilon}:\mathbb{R}\rightarrow\mathbb{C}, \gamma_{\varepsilon}(t)=\begin{cases}-1-t+\varepsilon i&t\le-1\\\varepsilon\exp(\pi i(1+\frac{t}{2}))&-...
2
votes
1answer
38 views

Calculate $\zeta(s) = \frac{1}{\Gamma(s)} \sum_{n=0}^\infty \frac{B_n}{n!} \int_0^\infty x^{s + n - 2} dx$

By Taylor expanding $$\frac{x}{e^{x}-1} = \sum_{n=0}^\infty \frac{B_n}{n!}x^n$$ in the Zeta function $$\zeta(s) = \frac{1}{\Gamma(s)} \int_0^\infty x^{s-2} (\frac{x}{e^{x}-1})dx$$ we find \...
0
votes
0answers
26 views

Pole Of Zeta function extension

Please advise on how to proceed with this? I don't know where to begin? Thanks. Show that the Riemann-Zeta function has a pole of order 1 at 1 after it has been extended to a holomorphic function on ...
1
vote
1answer
52 views

Deriving $ \frac{1}{n^s} = \frac{1}{\Gamma(s)} \int_0^{\infty}t^{s-1} e^{-nt} \, dt $ Backwards?

Is it possible to start with $\dfrac{1}{n^s}$ and then, without knowing the Gamma function in advance, naturally (with reasons!) derive that $$ \frac{1}{n^s} = \frac{1}{\Gamma(s)} \int_0^{\infty}t^{s-...
9
votes
2answers
312 views

Closed form for $ S(m) = \sum_{n=1}^\infty \frac{2^n \cdot n^m}{\binom{2n}n} $ for integer $m$?

What is the (simple) closed form for $\large \displaystyle S(m) = \sum_{n=1}^\infty \dfrac{2^n \cdot n^m}{\binom{2n}n} $ for integer $m$? Notation: $ \dbinom{2n}n $ denotes the central binomial ...
0
votes
0answers
22 views

A new formula relating the factorial and Riemann Zeta function resp. Bernoulli numbers?

I proved the following identities involving the factorial and Riemann's Zeta function respectively the Bernoulli numbers: $$\sum _{k=1}^{i}-{\frac {{\pi }^{-2\,k}\zeta \left( 2\,k \right)\left( -1 \...
0
votes
1answer
62 views

Is the given expression convergent as $n\to\infty$?

I want to know whether the following expression is convergent as $n\to\infty$ $$\frac{1}{n}\sum\limits_{k=1}^{\infty}\frac{|\ln n-\ln k|}{k^{(1+1/n)}}\cdot$$ With use of Riemann zeta function $\zeta(s)...
0
votes
0answers
93 views

Analytic Bound on The Riemann Zeta Function

Given the canonical infinite product representation (Weierstrass form) of the gamma function, $$\Gamma(z)= \left [ze^{\gamma z}\prod_{m=1}^{\infty} \left ( 1+ \frac{z}{m} \right)e^{-z/m} \right ]^{-1} ...
0
votes
0answers
113 views

Finding the singularities and residues of a Gamma/Riemann Zeta function.

The function I have is $f(z)=\zeta(z)\Gamma(z − 1)\sin(\pi z)$ and I need to find all singularities and their residues so I can evaluate a clockwise contour integral for the contour $\left\lvert z+\...
1
vote
1answer
56 views

Signature of The Riemann Zeta Function

Let $\zeta\colon\mathbb{C}\to\mathbb{C}$ denote the Riemann zeta function, analytically defined on $\mathbb{C}$ by meromorphic continuation, where $z \in D$. Similarly, let $G$ represent the co-domain ...
0
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0answers
31 views

Relationship between Riemann Zeta function and Prime zeta function

In his paper, Daniel Grunberg shows a relationship between the Stirling Numbers of the first kind and the Harmonic numbers via series of partitions (see Equation 3.1 on Page 5 in the link above). If ...
1
vote
1answer
35 views

On $-\frac{\zeta'(x)}{x\zeta(x)}$ and von Mangoldt function

I believe that it is possible show the following Fact. For real $x>e$ then $$-\frac{\zeta'(\log x)}{x\zeta(\log x)}=\sum_{n=1}^\infty\frac{\Lambda(n)}{n^{\log x}},$$ where $\zeta(x)$ is the ...
1
vote
1answer
70 views

Is it known if $\frac{\zeta(3)}{\pi^3}\in\mathbb{Q}$? [duplicate]

Is it known if $\frac{\zeta(3)}{\pi^3}\in\mathbb{Q}$? It is obvious that $\frac{\zeta(2n)}{\pi^{2n}}\in\mathbb{Q}$, but since there is no closed form for the odd values, are we left to be unable to ...
1
vote
1answer
109 views

The solution to $\sum_{n=1}^\infty\frac1{n^3}$ in a form not considered closed form?

Is there a solution to $\sum_{n=1}^\infty\frac1{n^3}$ that isn't in the standard closed form? I was wondering if a looser definition of "solution" could allow someone to solve this or if the solution ...
0
votes
1answer
60 views

The theory of riemann zeta function titchmarsh page 15 question in the proof of the functional equation

I am currently reading Titchmarsh's book about the Riemann Zeta function and came across a problem in a proof of the functional equation that I cannot solve. To be precise, I am referring to this ...
6
votes
0answers
129 views

A generalization of an integral related with $\zeta(2)$

It is pretty well-known (and not difficult to prove) that: $$ \int_{0}^{+\infty}\frac{x}{e^{x}-1}\,dx = \zeta(2) = \sum_{n\geq 1}\frac{1}{n^2} \tag{1}$$ but what is known about $$ I_2 = \int_{0}^...
2
votes
2answers
61 views

Analytic continuation of Riemann zeta-function

I showed that $$\zeta(s)=\frac1{1-2^{1-s}}\sum_{n=1}^\infty (-1)^{n-1}\frac1{n^s}.$$ Right hand side should be analytic when Re$(s)>0$ and $s\neq 1$, but there are problems at points $s=\frac{2k\...