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

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31
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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
126 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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0answers
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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( ...
7
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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
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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
393 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
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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?
5
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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
220 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 ...
3
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1answer
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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 $$ ...
0
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2answers
219 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
votes
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
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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
348 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
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2answers
155 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
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4answers
243 views

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

How can it be shown that the Riemann $\zeta(s)$ function has no zeroes for $\Re(s) > 1$?
10
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1answer
285 views

Prove the Wallis formula form $\left(4^{\zeta{(0)}} \cdot e^{-\zeta'{(0)}}\right)^2=\frac{\pi}{2}$

How would you prove the following Wallis formula form $$ \left(4^{\zeta{(0)}} \cdot e^{-\zeta'{(0)}}\right)^2=\frac{\pi}{2}?$$ Thanks in advance!
8
votes
2answers
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How to prove $\zeta'(0)/\zeta(0)=\log(2\pi)$?

How do I prove that $\zeta'(0)/\zeta(0)=\log(2\pi)$ ? I can get $\zeta(0)=-\frac{1}{2}$, but I don't know how to calculate $\zeta'(0)=-\frac{1}{2}\log(2\pi)$ ? Can you help me ? Here $\zeta(s)$ is ...
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2answers
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Sum of Stieltjes constants

Does anyone know of any papers or resources dealing with the following question: For which values of $s=\sigma+it$ does the following sum of Stieltjes constants hold, ...
2
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1answer
50 views

Dense Zeta Curve

In Reference 4 of this Wikipedia article, it is stated that the curve $\{(\zeta(\sigma+it),\zeta^{(1)}(\sigma+it), \cdots, \zeta^{(n-1)}(\sigma+it))|t\in\mathbb R\}$ is dense in $\mathbb C^n$ if ...
11
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1answer
336 views

Closed form for $\sum_{n=2}^\infty \frac{1}{n^2\log n}$

I had attempted to evaluate $$\int_2^\infty (\zeta(x)-1)\, dx \approx 0.605521788882$$ Upon writing out the zeta function as a sum, I got $$\int_2^\infty ...
7
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1answer
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Derivative of the Riemann zeta function for $Re(s)>0$.

The Riemann zeta function can be analytically continued to $Re(s)>0$ by the infinite sum $$\zeta(s)=\frac{1}{1-2^{1-s}}\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^s}.$$ Can we differentiate this with ...
3
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0answers
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A Hamiltonian with smooth term exact to the Riemann zeros

what would happen if one found a Hamiltonian with an smooth level density in the form $$ N(E)= \frac{E}{2\pi}\log\left(\frac{E}{2\pi e}\right)$$ which is exactly the density of the RIemann zeros.. ...
2
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1answer
182 views

Meaning of equality in zeta regularization

It is known that $$\sum\limits_{n = 1}^\infty{n = 1 + 2 + 3 + \cdots} = \infty$$ but it is also known that $$\sum\limits_{n = 1}^\infty{n = 1 + 2 + 3 + \cdots} = -\frac{1}{{12}}$$ which can obtained ...
3
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1answer
187 views

How to prove the identity $\pi^{s/2}=e^{(\log(2\pi)-1-\gamma/2)s}\prod_{\rho}e^{s/\rho} $?

In the wikipedia the Hadamard product for the Riemann's zeta function has two forms. The first one is ...
4
votes
2answers
465 views

Conditional convergence of Riemann's $\zeta$'s series

Do Riemann's zeta-function's partial sums $\sum_{n=1}^N n^{-s}$ converge conditionally for some value $s=\sigma+it$ with $\sigma\le 1$? (We must at least have $t\ne 0$ of course.) Partial summation ...
5
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0answers
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Zeta function values in terms of Bernoulli numbers.

The material presented at this link on Zeta function values at even integers proposes a method to compute these that is based on Euler's work. I would like to present a short proof for your ...
3
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1answer
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Identity for $\zeta(k- 1/2) \zeta(2k -1) / \zeta(4k -2)$?

Is there a nice identity known for $$\frac{\zeta(k- \tfrac{1}{2}) \zeta(2k -1)}{\zeta(4k -2)}?$$ (I'm dealing with half-integral $k$.) Equally, an identity for $$\frac{\zeta(s) \zeta(2s)}{\zeta(4s)}$$ ...
0
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1answer
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Riemann's Zeta function [duplicate]

Possible Duplicate: Riemann Zeta Function and Analytic Continuation Calculating the Zeroes of the Riemann-Zeta function It is stated that Riemann's Zeta function has zeros at negative ...
4
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1answer
152 views

Are these two facts related?

I was told some days ago that the possibility of two randomly picked numbers are relatively prime to each other is $6/(\pi^2)$. And it is well known that the value of Riemann zeta function at 2 is ...
30
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1answer
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A Bernoulli number identity and evaluating $\zeta$ at even integers

Sometime back I made a claim here that the proof for $\zeta(4)$ can be extended to all even numbers. I tried doing this but I face a stumbling block. Let me explain the problem in detail here. I was ...
3
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1answer
219 views

Is it possible to solve this equation analytically?

The following equation holds: \begin{align} & \frac{9}{2}\pi \\[8pt] & = x \\[8pt] & {}+\cos (x) \\[8pt] & {}+\cos (x+\cos (x)) \\[8pt] & {}+\cos (x+\cos (x)+\cos (x+\cos (x))) ...
3
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1answer
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Riemann Zeta Function Manipulation

The Riemann zeta function is defined on the $Re z> 1$ by $$\zeta(z)=\sum_{n=1}^\infty \frac{1}{n^z}$$ (i) show that for $Re z> 1$, we have $$(1-2^{1-z})\zeta(z)=\sum_{n=1}^\infty ...
3
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0answers
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How to get the floor function as a Mellin inverse of the Hadamard product of the Riemann zeta function?

The floor function is given - by Perron's formula - as a Mellin inverse of the zeta function. namely : $$\left \lfloor x \right \rfloor=\frac{1}{2\pi ...
3
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4answers
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Does Zeta converge for Complex numbers Re(s)>1 where imaginary part is not zero?

I know values of zeta for s= 2,3,4,... but what's the value of zeta as an example s= 2+14i
3
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2answers
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What is the explanation for similar decimal digits in values of Riemann zeta function with certain arguments close to one?

In Mathematica I tried these values close to one as arguments for the Riemann zeta function: ...
2
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0answers
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If RH is false , could this be true?

Let $\zeta(s)$ be the Riemann zeta function. Assume RH is false , is it possible that we have in the critical strip $\zeta(a_1+ti) = \zeta(a_2+ti) = \zeta(a_3+ti) = \cdots = \zeta(a_n+ti) = 0$ For ...
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1answer
120 views

Hurwitz zeta function

I am using the below function to compute the Hurwitz zeta function from Riemann zeta function. But I am not getting the correct results when compared with the value of Wolfram alpha Hurwitz zeta ...
1
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1answer
208 views

How does $\zeta(1 - s)$ become $(-1/s + \cdots)$?

Why is $$\zeta(1 - s) = -\frac{1}{s} + \cdots$$ for small negative values of $s$? A detailed explanation would be appreciated.
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Is this Fourier like transform equal to the Riemann zeta function?

This question builds upon the answer to this question. This new question has only minor changes compared to the previous question, but the scale factor of the output from the Fourier like transform is ...
2
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2answers
256 views

Sum of the Stieltjes constants? (divergent summation)

The sequence of Stieltjes-constants diverges and thus cannot be summed conventionally. However their signs oscillate (unfortunately non-periodic) and thus I tried Euler- and a version of ...
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0answers
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Approximation of distribution of $\pi_k(n)$ using $\zeta(s)$

Let $\pi_k(n) $ be the number of numbers with k prime factors (repetitions included) less than or equal to n. If we take the sums: $z_1(s) = \sum_{n= 1}^\infty \frac{1}{(p_{1,n})^s},~ z_2(s) = \sum ...
7
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1answer
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What is the formula for the first Riemann zeta zero?

I found this approximation of which an earlier version I posted in the chat room: $$7 \pi -\text{Log}\left[\frac{7}{2} e^{-7 \pi /2}+\frac{5}{2} e^{-5 \pi /2}+\frac{3}{2} e^{-3 \pi /2}+e^{5 \pi /2}+2 ...
1
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1answer
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Can this expression be simplified and is the real part of the left hand side equal to minus pi?

By starting with some Dirichlet series similar to logarithms, or somewhat similar to logarithm Dirichlet series, I arrived at this expression: ...
7
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1answer
245 views

Zeta functions of groups and an identity of Ramanujan

Zeta functions are being developed for all sorts of mathematical objects these days. One general situation is that of zeta functions of groups. If $G$ is a finitely generated group then we let ...
3
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2answers
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Elementary derivation of certian identites related to the Riemannian Zeta function and the Euler-Mascheroni Constant

Is the proof of these identities possible, only using elementary differential and integral calculus? If it is, can anyone direct me to the proofs? ( or give a hint for the solution ) ...
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2answers
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Detailed proof of $\zeta(s)-1/(s-1)$ extends holomorphically to $\Re(s)>0$

I'm trying to understand the proof of PNT by Don Zagier. But his proof is too simplified so I can't understand it. I got stumped at step II: $\zeta(s)-1/(s-1)$ extends holomorphically to ...
5
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3answers
807 views

Derivatives of the Riemann zeta function at $s=0$

It's a curious fact that for $n>0$, $\zeta^{(n)}(0)\approx -n!$. Apostol gave a table for $\frac{\zeta^{(n)}(0)}{n!}$, among other results on $\zeta^{(n)}(0)$ . the sequence : $$\delta_{n}=\left | ...
3
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2answers
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Modern formula for calculating Riemann Zeta Function [duplicate]

Possible Duplicate: How to evaluate Riemann Zeta function I have an amateur interest in the Zeta Function. I have read Edward's book on the topic, which is perhaps a little dated. I would ...