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

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Abel-Plana formula for $\zeta(s)$, is this integral approximation correct?

I wrote a computer program to calculate values for $\zeta(s)$. I was scanning for something that would calculate complex values for $\zeta(s)$. I found the following approximation under the Integral ...
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3answers
120 views

Closed form for $\sum_{k=1}^\infty(\zeta(4k+1)-1)$

Wikipedia gives $$\sum_{k=2}^\infty(\zeta(k)-1)=1,\quad\sum_{k=1}^\infty(\zeta(2k)-1)=\frac34,\quad\sum_{k=1}^\infty(\zeta(4k)-1)=\frac78-\frac\pi4\left(\frac{e^{2\pi}+1}{e^{2\pi}-1}\right)$$ from ...
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Can anyone improve on this work and find a closed form of $\zeta(3)$?

This was something I and another user came across independently, although he decided to post it on reddit. So while its already online, let me reproduce it here with the hope that someone will be able ...
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Non-standard numbers and exponential form of Zeta function

Basic idea For a long time I was looking for a numerical system that would allow to compare infinite sets. In contrast to Cantor's approach that empathizes the possibility of on-to-one correspondence ...
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Riemann zeta function Euler product for primes equivalent to $3$ mod $4$

Question: can $$ \zeta_1(s) = \prod_{p \equiv 3 \pmod{4}} \frac{1}{1 - p^{-s}} $$ be evaluated or written in terms of standard functions? Details: We can write the Riemann zeta function as ...
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2answers
68 views

Proof of Functional Equation Zeta

$$ \pi^{-s/2}\zeta(s)\Gamma(s/2)=\pi^{-(1-s)/2}\zeta(1-s)\Gamma((1-s)/2) $$ (That's the equation that want to prove) Hello guys, so I'm trying to prove the functional equation of Riemann Zeta, ...
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1answer
330 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 !?
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Weierstrass's M-test example for uniform convergence and switching Sum and Integral.

How would I go about finding $M_n$ in \begin{equation} \sum_{n=1}^{\infty} \int_{0}^\infty x^{\frac{s}{2}-1}e^{-\pi n^{2}x}dx \end{equation} to show that it is uniformly convergent? UPDATE: Just ...
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Is there anything known about the zeros of $\displaystyle \sum_{n=1}^{\infty} \left(\frac{1}{\rho_n^s} +\frac{1}{\overline{\rho_n}^s}\right)$?

Assuming the RH and $\rho_n =\frac12 + \gamma_n i$ being the n-th non-trivial zero of $\zeta(s)$, then numerical evidence suggests that: $$f(s) :=\displaystyle \sum_{n=1}^{\infty} ...
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1answer
45 views

Trivial zeroes of Zeta are simple

The trivial zeroes of the Riemann Zeta function are located on $-2\mathbb N^*$ and they are simple. It is not difficult to see that, but the proof I have in mind is using the fact that ...
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1answer
134 views

Infinite Series $\sum_{n=2}^{\infty}\frac{(-1)^n}{n^k}\zeta(n)$

I can prove that $$\displaystyle \sum_{n=2}^{\infty}\frac{(-1)^n}{n}\zeta(n)=\gamma$$ Is there any known value for $\displaystyle \sum_{n=2}^{\infty}\frac{(-1)^n}{n^k}\zeta(n)$ for any natural number ...
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Riemann zeta function, functional equation, what completes this analogy?

What completes this analogy? This: $$\zeta(s) = 2^s\pi^{s-1}\ \sin\left(\frac{\pi s}{2}\right)\ \Gamma(1-s)\ \zeta(1-s)\;\;\;\;\;\;\;\;\;\;(1)$$ is to: $$\chi(s)=\pi ^{-\frac{s}{2}} \Gamma ...
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1answer
89 views

regularization of sum $n \ln(n)$

I was testing out a few summation using my previous descriped methodes when i found an error in my reasoning. I'm really hoping someone could help me out. The function which i was evaluating was ...
8
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2answers
279 views

A double series $\frac13 \sum_{j=1}^{\infty}\sum_{i=1}^{\infty}\frac{(i-1)! (j-1)!}{(i+j)!}H_{i+j}$ giving $\zeta(3)$

Here is a symmetric rational double series giving Apery's constant: $$ \frac13 \sum_{j=1}^{\infty}\sum_{i=1}^{\infty} \displaystyle \frac{(i-1)! (j-1)!}{(i+j)!} H_{i+j} = \zeta(3) $$ where ...
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1answer
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Why is $\zeta(1+it) \neq 0$ equivalent to the prime number theorem?

Reading through Titchmarsh's book on the Riemann zeta function, chapter 3 discusses the Prime Number Theorem. One way to prove this result is to check the zeta function has no zeros on the line $z = ...
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5answers
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What is so interesting about the zeroes of the Riemann $\zeta$ function?

The Riemann $\zeta$ function plays a significant role in number theory and is defined by $$\zeta(s) = \sum_{n=0}^\infty \frac{1}{n^s} \qquad \text{ for } s > 1 \text{ and } s= \sigma + it$$ The ...
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1answer
58 views

How to manipulate this functions to an identity involving the Riemann zeta function

The identity I want to prove is the following (from Stein's book, an introduction to Fourier analysis): $$\pi^{-s/2} \Gamma(s/2) \zeta (s)=\frac{1}{2} \int_{0}^{\infty}t^{\frac{s}{2}-1}(v(t)-1)dt$$ ...
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1answer
183 views

Fibonacci numbers and the nontrivial zeros of the Riemann zeta function

Is this a mathematical coincidence? For $n=1,\dots,7$: $$ \left\lfloor \prod_{k=1}^n \arg\left(\rho_k\right)\right\rfloor = F_{n+1}, $$ where $\arg$ is the complex argument, $\rho_n$ is the $n$th ...
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2answers
332 views

From the series $\sum_{n=1}^{+ \infty} \left(H_{n}-\ln n-\gamma -\frac{1}{2n}\right)$ to $\zeta(\frac{1}{2}+it)$

Here is a pretty series $$ \displaystyle \sum_{n=1}^{+ \infty} \left(H_{n}-\ln n-\gamma -\frac{1}{2n}\right)=\frac{1}{2} \left(1-\ln (2\pi)+\gamma\right) \tag{*} $$ where $H_{n}:=\sum_{1}^{n} ...
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1answer
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Is the functional equation for $\zeta (s) \left(1-\frac{1}{3^{s-1}}\right)$ known?

It says in wikipedia that Hardy gave a simple proof of the functional equation for: $$\eta(s)=\zeta (s) \left(1-\frac{1}{2^{s-1}}\right)$$ and that it is: $$\eta(-s) = 2 ...
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Can this relation be made into a functional equation?

I am trying to find the functional equation for this: $$\zeta(s)\left(1 - \frac{1}{n^{(s - 1)}}\right)$$ Therefore I let: $$x_1=\left(1-\frac{1}{n^{s-1}}\right)$$ which I substitute with ...
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1answer
153 views

Another question from Titchmarsh's Riemann Zeta Function.

I want someone to clarify me the next step in page 11. They write for $\delta=\prod p^l$ ,$k=\prod p^{\lambda}$ the sum: $\sum_{\delta |k} ...
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1answer
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A question from Titchmarsh's The Theory of the Riemann Zeta-Function.

On pages 35-36 here, we have that the integral $$\frac{1}{2i\sqrt{y}}\int_{1/2-i\infty}^{1/2+i\infty}\phi(s-1/2)\phi(1/2-s)(s-1)\Gamma(1+s/2)\pi^{-s/2}\zeta(s)y^sds$$ equals for $\phi(s)=1$ to: ...
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3answers
920 views

$1 + 1 + 1 +\cdots = -\frac{1}{2}$

The formal series $$ \sum_{n=1}^\infty 1 = 1+1+1+\dots=-\frac{1}{2} $$ comes from the analytical continuation of the Riemann zeta function $\zeta (s)$ at $s=0$ and it is used in String Theory. I am ...
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Please help me understand Analytic Density $\lim_{\sigma \to 1^+}\frac{1}{\zeta(\sigma)}\sum_{n \in A} \frac{1}{n^{\sigma}}$

$d (A) = \lim_{\sigma \to 1^+}\frac{1}{\zeta(\sigma)}\sum_{n \in B} \frac{1}{n^{\sigma}}$ for $B \subset \Bbb{N}$. So clearly this limit is $0$ for reciprocally summable (convergent) $B$. My goal ...
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Question Relating Gamma Function to Riemann Zeta function evaluated at integers

I was just reading a paper of Ramanujan entitled "On question 330 of Professor Sanjana" when I got confused regarding a proposition which I am unable to answer. The proposition is: $ \displaystyle ...
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1answer
189 views

Expanding Riemann Zeta

Consider the Riemann Zeta Function $\zeta(x) = 1 + 2^{-x} + 3^{-x} + 4^{-x}...$ Notice the following identity: $a^{-x} = (e^{ln(a)})^{-x} = e^{-xln(a)}$ Therefore: $\zeta(x) = 1 + 2^{-x} + 3^{-x} ...
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Why does $1+2+3+\cdots = -\frac{1}{12}$?

$\displaystyle\sum_{n=1}^\infty \frac{1}{n^s}$ only converges to $\zeta(s)$ if $\text{Re}(s) > 1$. Why should analytically continuing to $\zeta(-1)$ give the right answer?
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Series with $\zeta$

How do I calculate the following series: $$ \zeta(2)+\zeta(3)+\zeta(4)+ \dots + \zeta(2013) + \zeta(2014) $$ All I know is that $\zeta(2)=\pi^2/6$ and $\zeta(4)=\pi^4/90$. But this is not enough to ...
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How to prove the Riemann hypothesis holds for the first non-trivial zero? [duplicate]

The Riemann hypothesis states that all non-trivial zeros of the Riemann zeta function $\zeta(z)$ lie on the critical line $\Re(z)=1/2$. The MathWorld page on this topic mentions that the hypothesis ...
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2answers
54 views

Abel's Summation formula help.

I want to be able to show that, \begin{equation} \sum_{p} \log(p) p^{-s} = s \int_{1}^{\infty} \frac{\theta(t)}{t^{s+1}} dt \end{equation} where $\theta(x) =\sum_{p \le x} \log p$. and $\theta(x) = ...
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The effect of roots of Dirichlet's $\beta$ function condenses to $\frac12\left(1+ie^{i2\pi\frac{p}4}\right)$

With the help of Raymond Manzoni and Greg Martin I was able to derive an explicit formula for the number of primes of the form $4n+3$ in terms of (sums of) sums of Riemann's $R$ functions over roots ...
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Riemann Zeta continued fraction approximants

In the paper Continued-Fraction Expansions for the Riemann Zeta Function and Polylogarithms by Djurdje Cvijovic and Jacek Klinowski, there is a claim that I cannot reproduce. In the abstract they ...
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220 views

Mean Value of a Multiplicative Function close to $n$ in Terms of the Zeta Function

Let $f(n)$ be a multiplicative function defined by $f(p^a)=p^{a-1}(p+1)$, where $p$ is a prime number. How could I obtain a formula for $$\sum_{n\leq x} f(n)$$ with error term $O(x\log{x})$ and ...
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1answer
117 views

How to prove that Riemann zeta function is zero for negative even numbers?

Can anyone please explain to me how to prove that Riemann zeta function is 0 for all negative even numbers. In many references , they have just given the statement without any proof. Any explanation ...
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3answers
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Trivial zeros of the zeta function

The trivial zeros of the Riemann zeta function are negative even integers. But I don't understand how that makes sense with the original definition of the function. $\zeta(-2) = \sum_{n=1}^{\infty} ...
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Increasing sequences and $\zeta$-type functions

The Riemann zeta function is defined as the sum $\zeta(s) = \sum_{n \geqslant 0} n^s$. The question is whether it globally characterizes the sequence of all natural numbers, in the following sense: ...
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Behaviour of $\zeta(s)$ near $s=1$

I would appreciate if somebody could run this over and see if it works out? any suggestions or pointers would be appreciated. I denote the standard eta function $\eta$ by $\zeta^{*}$. I have not used ...
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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 ...
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Which mathematical objects generate the zeroes of $L$-functions?

I've studied analytic and algebraic number theory for years and years, and I encountered a hard question about Riemann zeta function and other kinds of $L$-functions - which might be one of the most ...
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1answer
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Upper bound for $|\zeta'(s)|$ near the line $\sigma=1$, a detailed proof

In page 285 Apostol leaves as a reader's asigment the proof that $|\zeta'(s)|=O(\log^{2}t)$, this is for every $T>0$ there exists a positive constant $K$ (depending on T) such that ...
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how to prove an approximate to Riemann Xi function having only real zeros

I am searching for approximates to Riemann Xi function. Riemann $\Xi(z)$ function is related to Riemann $\zeta(s)$ function via $$\Xi(z)=\xi(1/2+iz)=\xi(s)=(1/2)s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)$$ ...
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An inequality for $|\zeta (s,a)|$, a detailed proof

In page 272 of [1], Apostol leaves as a reader's assigment to complete a proof of a related statement with Hurwitz zeta function, defined initially for $\sigma >1$ by the series ...
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2answers
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Weird thing about $\sum_{k=2}^{\infty}(-1)^k\zeta{(k)}$

Consider the sum $S=\sum_{k=2}^{\infty}(-1)^k\zeta{(k)}$. By a simple manipulation, we can show: $$ ...
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2answers
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$\zeta(2n)$ proof [duplicate]

Can anybody pass me on a good source to see the steps in proving, \begin{equation} \zeta(2n) = \frac{(-1)^{k-1}B_2k (2 \pi)^{2k}}{2(2k)!} \end{equation} I know how we start by looking at the product ...
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1answer
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Why is the Bernoulli Number $B_1$ sometimes $+ \frac{1}{2} $?

By using the recursive formula, \begin{equation} \sum_{i=0}^{n} \binom{n+1}{i} B_i = n+1 \end{equation} we find $B_1$ to be $\frac{1}{2}$ and not $- \frac{1}{2}$. Why is this?
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Looking for a way to apply the Taylor Series expansion to find derivatives for a function.

This post references the Riemann-Siegel formula found at here and at here. I am writing a Java program which implements this formula. I am having trouble with the remainder terms. The Riemann-Siegel ...
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2answers
846 views

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 ...
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1answer
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Simple Zero of the Riemann Zeta Function

Let $s=σ+it$. Assume that $ζ(s)-1/(s-1)$ has an analytic continuation to the half plane $σ>0$. Show that if $s = 1 + it$, with $t≠0$, and $ζ(s) = 0$ then $s$ is at most a simple zero of $ζ$. I ...