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

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1answer
70 views

What does this limit indicate?

$$\lim_{x\rightarrow\infty} \zeta(x)-\zeta(x)^{-1}-\zeta(x)^2 = -1$$ What does this limit indicate?
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1answer
117 views

What's the lowest real $x$ such that $\zeta(x)$ converges?

It's easy to prove that$$\zeta(1)=\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+...$$ diverges, and $$\zeta(2)=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+...$$ converges to $\frac{\pi^2}{6}$. Intuiting the ...
12
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5answers
387 views

Alternating sum of multiple zetas equals always 1?

This is more in the category of "recreational math"... I was playing with multiple zetas, in the notation of $\zeta(k),\zeta(k,k),\zeta(k,k,k),\ldots$ as given in wikipedia. Looking at the alternating ...
-2
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3answers
220 views

Quantum uncertainty can explain the Riemann Hypothesis?

In the recent paper "Riemann Hypothesis as an Uncertainty Relation" (http://arxiv.org/abs/1304.2435) the author claims that the presence of zeros out of the critical line may lead to the violation of ...
0
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1answer
198 views

Can anyone provide a proof for this conjecture?

Theorem?: Let $n$ be a positive interger, $n>1$, then Riemann zeta function can be expressed in terms of a multiple integral which exhibits the following form: $$ \displaystyle ...
15
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1answer
366 views

A Tough Series $\sum_{k=1}^\infty \frac{\zeta(2k+1)-1}{k+1}=-\gamma+\log(2)$

I have done series with $\zeta(2k)$ and $\zeta(k)$, but I have no idea with this one: $$\sum_{k=1}^\infty \frac{\zeta(2k+1)-1}{k+1}=-\gamma+\log(2)$$ $\gamma$ is the Euler–Mascheroni Constant. This ...
8
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4answers
432 views

Sum : $\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)^3}$

Prove that : $$\sum_{k=0}^{\infty} \frac{(-1)^{k}}{(2k+1)^3}=\frac{\pi^3}{32}.$$ I think this is known (see here), I appreciate any hint or link for the solution (or the full solution).
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1answer
52 views

The minimum of a function

Could anyone possibly give me any help with finding the minimum of this function? I believe the result to be $2\pi |n|$ from page 619 of this paper by W. G. C. Boyd. \begin{equation} ...
1
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1answer
202 views

How does one calculate the amount of time required for computation?

For example, to compute the zeroes of the Riemann zeta function using the Euler-Maclaurin summation method one has to do O(T) work. The Euler-Maclaurin summation method for zeta is given by $ ...
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2answers
173 views

Riemann Zeta Function By Hand

This may be a stupid question but is there a way to calculate Riemann's Zeta Function by hand exactly or can you only estimate it?
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1answer
464 views

The meaning of the Euler Formula for Zeta

Does anybody know about a "meaning" behind the Euler Formula, what does it really say about the primes? I know that it is in equation to the zeta function and also how it is derived, but cannot find ...
2
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1answer
229 views

1+2+3+4+… = -1/12 [duplicate]

Consider the zeta function $\zeta(s)= \sum \limits_{n=1}^{\infty} \frac{1}{n^s}$. It is established that $ \zeta(-1) = -\frac{1}{12}$. Reference (Equation 90) Then we have $ \zeta(-1) = \sum ...
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0answers
40 views

The sum of the reciprocals of fourth powers [duplicate]

This problem is an extension of the well known basel problem and involves finding the sum of 1 + 1/16 + 1/81 ... = 1/1^4 + 1/2^4 + 1/3^4 ... 1/n^4 where n tends to infinity Euler managed to prove ...
3
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1answer
186 views

Discontinuities of $\sum \frac{x^{\rho}}{\rho}$

H. Edwards in his book on the zeta function says that $\sum\frac{x^{\rho}}{\rho}$ converges conditionally "even when $\rho ,1-\rho$ are paired." I tried calculating some terms (n = 500 or so) and ...
5
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3answers
129 views

Two trivial questions about zeta function

I have two questions concerning the Riemann zeta function which are rather trivial so if anyone can give me the answers that would be nice, here is what I`m interested in: 1) In the equality ...
18
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1answer
357 views

References to integrals of the form $\int_{0}^{1} \left( \frac{1}{\log x}+\frac{1}{1-x} \right)^{m} \, dx$

While extending my calculation techniques, with aid of Mathematica, I found that \begin{align*} \int_{0}^{1}\left( \frac{1}{\log x} + \frac{1}{1-x} \right)^{3} \, dx &= -6 \zeta '(-1) ...
2
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2answers
182 views

Series expansion of $\zeta(s)$ using the derivatives of $\zeta(0)$

When playing with series expansions, I stumbled upon the following relation: $$\zeta(s) := \sum \limits_{n=0}^{\infty} \left( \zeta^{(n)}(0) \frac{s^n}{\Gamma(n+1)} \right)$$ that seems to hold for ...
3
votes
1answer
144 views

Find the value of: $ J=\int_{0}^{\infty}\frac{x^3}{e^x-1}\ln(x)\,dx $

I'm trying to find to value of: $$ J=\int_{0}^{\infty}\frac{x^3}{e^x-1}\ln(x)\,dx $$ Here's what I've done: $$ ...
8
votes
2answers
353 views

How to find integral of $\int_0^\infty \frac{\ln ^2z} {1+z^2}\mathrm{d}z$?

How do I find the value of $$\int_{0}^{\infty} \frac{(\ln z)^2}{1+z^2}\mathrm{d}z$$ without using contour integration, - using the usual special functions, e.g., zeta/gamma/beta/etc. Thank you,
12
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2answers
407 views

Can the Basel problem be solved by Leibniz today?

It is well known that Leibniz derived the series $$\begin{align} \frac{\pi}{4}&=\sum_{i=0}^\infty \frac{(-1)^i}{2i+1},\tag{1} \end{align}$$ but apparently he did not prove that $$\begin{align} ...
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0answers
110 views

Efficient way for computation of derivatives of $f(x) = \zeta(1-x) + 1/x $ at integer x?

For some exercises with (divergent) summation of the Stieltjes constants I'm trying a formula, which involves derivatives of the $\zeta()$ -function at negative integers; perhaps better formulated as ...
8
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1answer
717 views

how to understand $\log\zeta(s)$ (Riemann zeta function)?

I know that if a function $f$ is analytic and has no zeros in a simple connected region, then we can define $\log{f}$ making it analytic in that region. Let's assume $Re(s)>1$. Is $\zeta(s)$ ...
3
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1answer
112 views

Calculating an almost Gamma integral

How would you proof that $$I:=\int_{0}^{\infty}\frac{z^{x-1}}{e^{z}+1}dz=\left(1-2^{1-x}\right)\Gamma(x)\zeta(x)$$ I can rewrite the integral as ...
17
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3answers
591 views

A series involves harmonic number

How do we get a closed form for $$\sum_{n=1}^\infty\frac{H_n}{(2n+1)^2}$$
7
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1answer
190 views

Serre's proof that zeta function is meromorphic

I try to understand the proof of Chap. VI, n° 3.1, Prop. 10 in Serre's "A course in arithmetic" (page 70). The goal is to prove that zeta-function can be written as \begin{align*} ...
4
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1answer
203 views

About Euler's formula for Apery number

Euler's formula. $$\zeta(3)=\frac{\pi^2}{7}\left(1-4\sum_{m\ge 1}\frac{\zeta(2m)}{(2m+1)(2m+2)2^{2m}}\right)$$ I saw this formula in Wikipedia a few months ago. I have searched about Euler's ...
9
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2answers
379 views

A tough series: $\sum_{k=1}^{\infty}\frac{\zeta(2k)}{(k+1)(2k+1)}$, need help

I was doing a integral which ends up with a tough series part: $$\sum_{k=1}^{\infty}\frac{\zeta(2k)}{(k+1)(2k+1)}$$ Mathematica says $$\frac12$$ Which agrees with the anwer...Anyone know how to ...
3
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1answer
98 views

How to evaluate $\xi(0)$?

How do I evaluate $\xi(0)$ for the Riemann xi function? I know $\xi(0) = \xi(1)$ and $\xi(0) = \tfrac{1}{2} \cdot 0 \cdot (-1) \cdot \Gamma(0) \cdot \zeta(0)$ $\xi(1) = \tfrac{1}{2} \cdot 1 \cdot 0 ...
3
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2answers
167 views

Another improper integral

Show that : $$\int_0^1\frac{(\sin ^{-1}x)^2}{x}\text{d}x=\frac{\pi ^2\ln 2}{4}-\frac78\zeta(3)$$ This integral is in "irresistible integrals" on page 122. I can't prove this one.
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2answers
296 views

Intuitive explanation with rigorous details why zeta has infinitely many zeros?

I have seen a proof outline that $\zeta$ has infinitely many zeros on the critical line here but what I really want is: Simplest possible (least "magic") argument that explains why zeta has ...
8
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3answers
940 views

Why does zeta have infinitely many zeros in the critical strip?

I want a simple proof that $\zeta$ has infinitely many zeros in the critical strip. The function $$\xi(s) = \frac{1}{2} s (s-1) \pi^{\tfrac{s}{2}} \Gamma(\tfrac{s}{2})\zeta(s)$$ has exactly the ...
5
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0answers
217 views

On the series $ \displaystyle \sum_{n=1}^{\infty} \frac{\Lambda(n)}{\sqrt{n}} \cos(x \log(n) + a) $.

Is the following series ‘summable’ in the sense that it may be divergent but we can attach a meaning to it? $$ \sum_{n=1}^{\infty} \frac{\Lambda(n)}{\sqrt{n}} \cos(x \log(n) + a) = (\text{reg}) ...
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0answers
125 views

Why should the eigenvalues of random matrices reflect zeta function zeroes?

As per this article. And also : Is this a particular property of 2 dimensional objects? Could random vectors also model the "universality phenomena" -- globally random distribution of zeroes combined ...
31
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2answers
3k views

Books about the Riemann Hypothesis

I hope this question is appropriate for this forum. I am compiling a list of all books about the Riemann Hypothesis and Riemann's Zeta Function. The following are exluded: Books by mathematical ...
1
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1answer
130 views

asymptotic growth of zeta function on the real line

let $r$ be real $r> 1$ then $$\zeta(r) = O\left(1+\frac{1}{r-1}\right)$$ Can you tell me how to prove this formula? I found this after posting $$\sum_{n \ge 1} \frac{1}{n^r} = 1 + \sum_{n \ge ...
2
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1answer
117 views

Theta series and Riemann Hypothesis

in the paper http://www.fuchs-braun.com/media/dd209bf5c2203a87ffff80a3ffffffef.pdf section 2 ' Hilbert-Polya space' page: 180 the author introduce the Theta series $$ F(\phi(x))= ...
2
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0answers
203 views

Proof of Euler's general formula for a sum involving harmonic numbers [duplicate]

I have seen this formula, but how to prove this? $$2\sum\limits_{k=1}^\infty \frac{H_k}{\left( k+1 \right)^m} =m\zeta \left( m+1 \right)-\sum\limits_{k=1}^{m-2}{\zeta \left( m-k \right)\zeta \left( ...
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0answers
289 views

How does one prove that $\zeta(3)$ is irrational?

How does one prove that $\zeta(3)$ is irrational ? I would like to know how Apery did it. In particular how a recursion gives rise to irrationality !?
2
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2answers
174 views

Estimating the integrated Tchebychev function and calculating its error

I would like to understand how to derive (2) from (1) below. Problem: If $\psi_1$ is the integrated Tchebychev function below $$\psi_1(x)=\frac{1}{2\pi i} ...
3
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0answers
400 views

An Expression for $\log\zeta(ns)$ derived from the Limit of the truncated Prime $\zeta$ Function

I think, here, I found $$ P_\color{red}x(\color{blue}s)=\sum_{p<\color{red}x} \frac{1}{p^{\color{blue}s}} =\sum_{\color{green}n=1}^{\infty}\frac{ \mu (\color{green}n)}{\color{green}n} ...
11
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2answers
823 views

How would proving or disproving the Twin Prime Conjecture affect proving or disproving the Riemann Hypothesis if at all?

How would proving or disproving the Twin Prime Conjecture affect proving or disproving the Riemann Hypothesis? What are the connections between both conjectures if any?
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0answers
254 views

Estimation for logarithm of Riemann zeta function

Let $\sigma >1-\dfrac{c}{2\log(|t|+3)},|t|>7/8,$ where $c$ is constant from Theorem about region without zeros of Riemann zeta function. Using the fact that $$\log \zeta(s) - \log \zeta(s_1) = ...
2
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1answer
224 views

Inequality for Riemann zeta function

Prove that for $1<\sigma_1 \leq \sigma_2$ is true that $$\frac{\zeta(\sigma_2)}{\zeta(\sigma_1)} \leq \left|\frac{\zeta(\sigma_2+it)}{\zeta(\sigma_1+it)}\right|\leq ...
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1answer
109 views

Sums of the negative integer powers of $\zeta$ zeros have an analytical expression…?

The Mathworks page on Riemann's $\zeta$ function says: Let $\rho_k$ denote the $k$th nontrivial zero of $\zeta(s)$, and write the sums of the negative integer powers of such zeros as $$ ...
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2answers
244 views

How one can obtain roots at the negative even integers of the Zeta function?

The Riemann zeta function $ζ(s)$ is defined for all complex numbers $s ≠ 1$ with a simple pole at $s = 1$. It has zeros at the negative even integers, i.e., at $s = −2, −4, −6, ...$. My question: ...
3
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2answers
162 views

Can the $\Xi(t)$ function be extended this way?

In 1893 Hadamard proved that: $$\xi(s) = \xi(0) \prod_{\rho} \left(1- \frac{s}{\rho} \right) \left(1- \frac{s}{1-\rho} \right)$$ where $\xi(z) = \frac12 z(z-1) \pi^{-\frac{z}{2}} \Gamma(\frac{z}{2}) ...
23
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3answers
2k views

Two Representations of the Prime Counting Function

The bounty for the best work out of Greg's answer, especially the "solving for $\pi^*(x;q,a)$ in terms of all $\Pi^*$ functions (tedious but possible)" part is over. Since Raymond's ...
3
votes
2answers
350 views

argument of the Riemann zeta function

what does it mean that the function $ F(t) $ $$ F(t)= \frac{\arg\zeta (\frac{1}{2}+it)}{\sqrt{\log\log(t)}} $$ is distributed as a 'Gaussian Random variable ?? in the limit $ t \to \infty $ a) $$ ...
4
votes
2answers
158 views

Simple Formula to Describe $\zeta$ in One Go

I was looking around for a simple formula that can describe Riemann Zeta $\zeta$ in one go, at $\mathbb C-\{1\}$, but I couldn't really find one. Could someone help me find one? I know one way to do ...
5
votes
4answers
245 views

Question about Riemann $\zeta(s)$ function zeroes

How can it be shown that the Riemann $\zeta(s)$ function has no zeroes for $\Re(s) > 1$?