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

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Question about the Riemann Zeta Function Proof [on hold]

The Riemann Hypothesis Theorem states that: There are infinitely many nontrivial zeros on the critical line and all these zeros have real part $\frac{1}{2}$. The proof is given by: $$\prod^\...
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0answers
102 views

Any underlying reason why these equations look similar? [on hold]

Questions Is there any way to go from either of these equations to the other? Or is there any more fundamental reason for their similarities? $$ \frac{1}{\zeta(s)} = \sum_{r=1}^\infty \frac{\mu(r)}{...
1
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2answers
66 views

Inequality involving sum of logarithms and hidden zeta-function

I would like to prove the following estimation: if $n \ge 2$ is a natural number, then $$\sum_{k=2}^n \frac{\log^2 k}{k^2} <2 - \frac{\log^2 n}{n}.$$ I have noticed that LHS is indeed bounded by ...
2
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1answer
34 views

Laplace transform of bell-shaped functions

A real smooth function $\varphi$ is said bell shaped iff as the Gaussian : $\varphi''$ is positive on $(-\infty,a) \cup (b,+\infty)$ and negative on $(a,b)$. I'm interested in the bilateral Laplace ...
3
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1answer
54 views

Prime Number Theorem and the Riemann Zeta Function

Let $$\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}$$ be the Riemann zeta function. The fact that we can analytically extend this to all of $\mathbb{C}$ and can find a zero free region to the left of the ...
4
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1answer
73 views

(Non-)Canonicity of using zeta function to assign values to divergent series

This article http://blogs.scientificamerican.com/roots-of-unity/does-123-really-equal-112/ got me thinking about the "identity" $$1 + 2 + 3 + \cdots = -1/12,$$ and I wanted to convince myself there ...
0
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1answer
32 views

Prove that this limit is the logarithmic derivative of the Riemann zeta function.

Prove the following limit: $$-\frac{\zeta '(s)}{\zeta (s)}=\lim_{c\to 1} \, \left(\frac{\zeta (c) \zeta (s)}{\zeta (c+s-1)}-\zeta (c)\right)$$ As a starting point I tried to enter this series ...
3
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4answers
112 views

Evaluating series of zeta values like $\sum_{k=1}^{\infty} \frac{\zeta(2k)}{k16^{k}}=\ln(\pi)-\frac{3}{2}\ln(2) $

Somehow I derived these values a few years ago but I forgot how. It cannot be very hard (certainly doesn't require "advanced" knowledge) but I just don't know where to start. Here are the sums: $$ \...
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0answers
50 views

Riemann's zeta function in a definite integral

I am trying to evaluate the value of the definite integral $ I= \int_0^1((1-\delta)\log[p]-\delta\log[1-p])^4dp$ Using Binomial expansion I get $I=(1-\delta)^4\int_0^1(\log[p])^4dp-4(1-\delta)^3\...
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1answer
45 views

What is the equivalent statement of GRH in term of Redheffer Matrix or Farey Sequences?

We all know that Riemann Hypothesis (RH) has many equivalent statements. There is one statement which expresses RH in term of Redheffer matrix, there is another equivalent statement of RH which ...
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1answer
74 views

How was the integral for Zeta Function created

How was the zeta function integrated from $$\zeta(s) = \sum_{n=1}^{\infty}\frac{1}{n^{s}}$$ To $$\zeta(s) = \frac{1}{\Gamma (s)}\int_{0}^{\infty}\frac{x^{s-1}}{e^{x}-1}dx$$ I've tried googling ...
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0answers
35 views

Is there any product formula for local zeta function?

Suppose that $V$ is a non-singular $n$-dimensional projective algebraic variety over the field $\mathbb{F}_q$ with $q$ elements. The local zeta function $Z(V, s)$ of $V$ (sometimes called the ...
0
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1answer
25 views

Is the functional equation for the inverse of Riemann zeta function valid for any point in the critical strip?

It is known that in the critical strip $s\in \{0<\mathrm{Re}(s)<1\}$,Riemann zeta function satisfies the following functional equation: $$\zeta(s)=\chi(s)\zeta(1-s),\tag{1}$$ $$\chi(s)=\frac{\...
1
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1answer
67 views

Riemann Zeta Function, Stirling's Numbers, and Infinite Series

A while back I was able to prove the following identity, $$\sum_{k=1}^{\infty}\frac{\Gamma(k+r)}{\Gamma(k)(k+r)^s}=\sum_{k=0}^{r}s(r+1,r+1-k)\zeta(s-r+k)$$ where $s(k,n)$ are the Sterling numbers of ...
1
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1answer
59 views

Is $\zeta(2n+1)\notin (2\pi)^{2n+1}\mathbb{Q}$ already known?

Is it already shown or at least conjectured that $$\zeta(2n+1)\notin (2\pi)^{2n+1}\mathbb{Q}?$$ You have any names and years who proved or conjectured it?
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2answers
46 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
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0answers
50 views

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

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
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1answer
71 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
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1answer
56 views

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

Does anybody know when Euler found his famous formula $$\zeta(2n)=\frac{(-1)^{n-1}(2\pi)^{2n}B_{2n}}{2(2n)!}?$$
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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(...
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1answer
59 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)^{-...
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0answers
36 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
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2answers
92 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^{-...
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0answers
64 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(...
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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
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1answer
91 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
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51 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 ...
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108 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
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3answers
105 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}...
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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?
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26 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 ...
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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
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1answer
76 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 ...
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25 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)=...
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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 ...
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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. ...
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43 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 ...
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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 ...
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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+\...
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2answers
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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 ...
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1answer
48 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 ...
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1answer
59 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 $\...
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1answer
234 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 ...
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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
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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
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4answers
213 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 ...
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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 ...
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2answers
45 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
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0answers
23 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
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1answer
39 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 \...