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

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How the steps followed help me to understand

Few days before I posted one problem on MSE here in which it was asked to determine the residue. I understood almost entire but got stuck the following Note that after step 4, in step 5 it is ...
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20 views

What is the Möbius analoge for Ihara's $\zeta$ function?

The Dirichlet series that generates the Möbius function is the (multiplicative) inverse of the Riemann zeta function; if s is a complex number with real part larger than 1 we have $$ ...
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42 views

Probabilistic interpretation for representation of unity using the zeta function

There's a cute identity, I believe due to Borwein, Bradley and Crandall (Section 4): $$1=\sum_{n=2}^\infty (\zeta(n)-1).$$ There are some generalizations in the linked paper as well. Question: Is ...
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3answers
48 views

Does the sum of the Zeta function taken on natural numbers converge?

Does this series \begin{equation*} \sum_{n\ge2} \zeta(n) \end{equation*} converge? If yes is it easy to prove ?
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27 views

On Zero-Free Regions for $\zeta(s)$ and $L(s,\chi)$ with $|t| \le 2$

I'm reading the proof from Hildebrand that for some $c_1 > 0$, the Riemann zeta function $\zeta(s)$ has no zero in the region $\sigma > 1-c_1$, $|t| \le 2$. (Here $s = \sigma + it$ per ...
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1answer
63 views

Find Values of Riemman Zeta Function [on hold]

How can I find all the possible values to the riemman zeta function? Is it possible to do so?
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1answer
83 views

Is there any relationship between the Riemann z function and strange attractors?

I have this question in mind since the first time I saw a graphical representation of the zeta function (like in the sample below). Just by looking to them I wondered if there is any relationship ...
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2answers
25 views

WHat is the inner steps here?

I was studying abour Riemann zeta function over here where in (3) it has been written "by Abel's theorem", we have $$\sum\limits_{n\geq 1}\frac{1}{n^s}=\sum\limits_{n\geq ...
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1answer
31 views

On the determination of residue

I need your help on the following. (1)First we are to find the residue of $\frac{x^s}{s}$ at $s=0$. Since $s=0$ is the pole of order 1, so we get Res$(\frac{x^s}{s},s=0)=\frac{1}{2\pi ...
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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 ...
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2answers
38 views

Summation with Riemann Zeta Function

So the Riemann zeta function $\zeta(s)$ is commonly defined as $\sum \limits_{n=1}^{\infty} n^{-s}$ Now, suppose that $a_k=\zeta (2k).$ How can I find the value of $$\sum \limits_{k=1}^{\infty ...
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1answer
97 views

Evaluate $\sum \sum 1/n^k $

I wanted to evaluate the sum: $$ \sum_{n \ge 2} \left(\zeta(n) - 1\right) $$ I rewrote this as: $$ \sum_{n\ge 2} \sum_{k\ge 2} \frac{1}{n^k} $$ I tried exploiting the symmetry but that didn't seem ...
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2answers
64 views

How to calculate $\zeta(i)$?

As the title says, I'm interested to know how $\zeta(i)$ is calculated. I know the functional equation for the zeta function, but if I put it in that in there, I must know $\zeta(1-i)$. Is it a good ...
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36 views

Provide me notes on Riemann zeta function to boast my knowledge to use in Research on Analytical Number Theory

I need your help. I want to study the Riemann zeta function from the very basic level, its concepts, theorems, solved problems etc. I am assigned one problem from Analytical Number Theory related to ...
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1answer
21 views

How the prove that the Dirichlet series converges?

I am ready to calculate an alternated series of the type of Dirichlet $L$-function. The series, as it is, is $$ L = ...
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1answer
73 views

Proving the Riemann Hypothesis and Impact on Cryptography

I was talking with a friend last night, and she raised the topic of the Clay Millennium Prize problems. I mentioned that my "favorite" problem is the Riemann Hypothesis; I explained what it posits ...
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1answer
25 views

Hunting for some properties of a series similar to Riemann zeta function.

I happened to meat a alternated zeta function in D. Borwein's paper(see eq. 40 of his paper). The series is $$ L_{-3}(s)=1-\frac{1}{2^s}+\frac{1}{4^s}-\frac{1}{5^s}+\frac{1}{7^s}-\frac{1}{8^s}+\dots ...
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1answer
42 views

Errors in following paper

In the paper "On the Level Curves of the Xi Function" http://arxiv.org/abs/1002.0352v8, John Breslaw takes a very similar approach to a study of the Riemann hypothesis I did while I was in my first ...
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25 views

Does this 'alternating' series with $\Lambda(n)$ converge for all $\Re(s)>0$?

The following equation is well known and valid for $\Re(s)>1$: $$\log\big(\zeta(s)\big)=\sum_{n=2}^\infty \dfrac{\Lambda(n)}{\log(n)\,n^s}$$ where $\Lambda(n)$ is the Von Mangoldt function. Take ...
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1answer
31 views

Elementary proof of an estimate on $\zeta$

I saw somewhere that If $s=\sigma +it$ where $\sigma >0$ and $t\in \mathbb R$,$x\geq |t|/\pi\implies \zeta(s)=\displaystyle \sum_{n\leq x} \frac{1}{n^s}+\frac{x^{1-s}}{s-1}+O(x^{-\sigma})$ ...
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2answers
58 views

Show that $\sum_{n \le x} \phi (n)=\frac{x^2}{2\zeta(2)}+ O(x \log x)$

How do I show that $\sum_{n \le x} \phi (n)=\frac{x^2}{2\zeta(2)}+ O(x \log x)$, where $O$ denotes the big-$O$ notation. And we already know that $\phi (n) = \sum_{d|n} \mu (d) \frac{n}{d}$. I ...
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1answer
110 views

Show that $1/\zeta(2k) = \sum_{m \le K} \mu (m)/m^{2k} + O(1/K)$

Show that $1/\zeta(2k) = \sum_{m \le K} \mu (m)/m^{2k} + O(1/K)$. I have already proved that $1/\zeta(s) = \sum_{m=1}^{\infty} \mu (m)/m^s$. But how do I show that if $k\ge 1$, $1/\zeta (2k) = ...
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211 views

Proof of $\zeta(2)=\frac{\pi^2}{6}$

While messing around with some integrals, I have found the following proof for $\zeta(2)=\frac{\pi^2}{6}$, but I'm not sure if it is valid: We take a look at the integral $I=\int_0^{\frac{\pi}{2}} ...
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23 views

Series of Riemann Zeta values analytic continuation.

What is the value of the series?: $$\sum_{k=1}^\infty\frac {\zeta(-k)} k$$ Where $\zeta(z)$ is the Riemann Zeta function and for every negative integer $n$ we have $\zeta(n)=-\frac {B_{n+1}} {n+1}$. ...
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48 views

Zeta zeros standard normal distribution

Below is a partially scaled plot of $\vartheta (\gamma_n) - \pi (n + 3/2) ,$ where $\gamma_n$ is the imaginary part of the $n$th zeta zero, and $\vartheta $ is the Riemann-Siegel theta function, for ...
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2answers
34 views

Limit in combination with an infinite series

How would I go about showing the following limits that involve infinite series $$ \lim_{x \to 0^{+}} \sum_{n=1}^{\infty} \frac{(-1)^n}{n^{2k+1}} \sin (2\pi n(x - \frac{1}{2})) = 0 \text{ with } k \in ...
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Analogue of Dirichlet $L$-function for $\mathbb{F}_q[T]$.

We consider an analogue of the Dirichlet $L$-function in $\mathbb{F}_q[T]$. Let $g \in \mathbb{F}_q[T]$, $g \neq 0$, let $\chi: (\mathbb{F}_q[T]/(g))^\times \to \mathbb{C}^\times$ be a homomorphism, ...
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1answer
27 views

Bounding Riemann zeta function by Euler product formula for finite $N$

In a paper concerning a quick calculation of Bernoulli numbers the following inequality is presented (page 3), only referring to it as "not hard to see" that it holds. $$ \sum_{n \leq N}^{} n^{-s} ...
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4answers
83 views

$\sum\limits _{n=1}^{\infty}\frac{1}{n^s}=50$ Riemann-Zeta

I've to find a value for 's' were the infinit sum gives me the value 50. Is that possible and how do I've to calculate that value. I've no idea how te begin so, help me! Solve s for: $$\sum\limits ...
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1answer
50 views

Is $f(z)=\sum_{k=1}^{\infty}\frac{1}{(k+\frac{1}{k})^{z}}$ somehow related to Riemann's zeta function?

I was looking at this series $$ f(z)=\sum_{k=1}^{\infty}\frac{1}{(k+\frac{1}{k})^{z}} $$ and wondering if it is somehow realted to the Riemann's zeta function $$ ...
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Is there anything known about the value where the Euler and Hadamard products for $\zeta(s)$ are equal?

Take the Hadamard product for the Riemann $\xi$-function ($\rho$ is a non-trivial zero of $\zeta(s)$): $$\xi(s) =\frac12\, s\,(s-1) \,\pi^{-\frac{s}{2}}\, \Gamma\left(\frac{s}{2}\right)\, \zeta(s) ...
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3answers
54 views

Integral equal to Riemann Zeta Function

As part of a homework problem in Rudin, I need calculate $$ \int_{1}^{N} \frac{[x]}{x^{s+1}} \,dx$$ where $[x]$ is the floor function. Clearly $[x]$ has derivative $0$ everywhere but the integers. ...
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1answer
42 views

What are 'Regular Products'?

When looking at the functional equation for the Riemann zeta function, I came across the statement: For $s$ an even positive integer, the product $\sin{(\frac{\pi s}{2})}\Gamma({1-s})$ is regular. ...
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54 views

Wrong proof of the functional equation for $ \zeta (s) $ but why is the result correct?

If I introduce the function $ f(x)= |x|^{s-1} $ inside Poisson summatory formula and use the fact that $$ \sum_{n=-\infty}^{\infty}|n|^{s-1}=2\zeta (1-s) $$ If I combine this expression in the ...
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1answer
53 views

How could I get access to more than the first 2 mln non-trivial zeros of $\zeta(s)$?

I would like to test whether or not the following product (or its complement) $$\displaystyle \displaystyle \prod_{n=1}^\infty \left(1- \frac{s}{\frac12+ (-1)^n\, \gamma_n \, i} \right)$$ converges ...
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2answers
46 views

Express the Riemann zeta function as the series with the divisor function $d(n)$ [closed]

How do I show this? $$\zeta(s)^2 = \sum_{n=1}^\infty \frac{d(n)}{n^s}$$ Here, $\zeta(s)$ is the Riemann zeta function, and $d(n)$ is the number of positive divisors of $n$. This looks like the ...
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28 views

Bounds for the imaginary part of the non-trivial zeros of the Riemann zeta function.

Let $\rho_{k}=\beta_{k}+i\gamma_{k}$ the $k-th$ non-trivial zero of the Riemann zeta funcion. We consider only the zeros with $\gamma_{k}>0$ . Then we have$$\gamma_{1}=14.13...$$ ...
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1answer
72 views

Non-Trivial Zeroes of Riemann Zeta Function

I am in the process of proving the prime number theorem, and I was wondering if there was any easy way of proving that $\sum_{\rho\in R} \rho^{-2}$ converges, where $R$ is the set of all non-trivial ...
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1answer
50 views

Some clarifications on analytic continuation of Riemann's Zeta function on $\frac 1 2$

Here's my problem: Riemann's Zeta function converge iff $x>1$ so if I want to have a finite value for $\zeta(\frac 1 2)$ I need to use it's analytic continuation but Riemann's hypothesis states ...
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54 views

A way to sum supernatural numbers involving Zeta function's analytic continuation

I have this idea on how to sum supernatural numbers assigning them a finite value in a way similar to how we assume that the sum of every natural numbers from 1 to infinity equals $-\frac 1 {12}$. ...
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1answer
25 views

Inequality $(n+1)^{-s} \leq (2n)^{-s}$ true for all $s\leq1$ and natural $n$?

On the line $S_{2n}-S_n$ I don't understand how the first inequality was established for $s \leq 1$. I see how it works for $0 \leq s \leq 1$ but not s < 0. Any clues?
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1answer
42 views

What does $O\left(\frac{1}{\log\log T}\right)$ mean?

I've started to work through this paper by Levinson (1975). The abstract says "Let $s = \sigma + it$. For any complex a, all but $O\left(\frac{1}{\log \log T}\right)$ of the roots of $\zeta(s) = a$ ...
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43 views

Taylor coefficients for $\zeta(x)\Gamma(x)$

Could you show that in a right neighbourhood of $x=0$ $$\frac{1}{n!}\frac{d^n}{dx^n}\Gamma(x)\zeta(x)$$ behaves like $(-1)^n \zeta (-1)+1$ when $n\to \infty$?
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Why does Titchmarsh say that we can move the derivative under $\frac{2}{\pi}\int_0^\infty \frac{\Xi(t)}{t^2 + \frac{1}{4}} \cosh(\alpha t) \, dt$

If we define the Riemann-Xi function as $$ \Xi(t) = \xi(\frac{1}{2} + it)$$ where $$\xi(s) = \frac{1}{2}s(s-1)\pi^{-\frac{s}{2}}\Gamma(\frac{s}{2})\zeta(s),$$ then according to Titchmarsh in his ...
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1answer
55 views

Explicit formula for $\zeta$

I can not find anywhere on the internet a proof of this formula : $$\sum_{p\in\mathbb{P}, \; m\geq 1, \; p^m < x} \ln p = x - \sum_\rho \frac{x^\rho}{\rho} - \ln 2\pi - \frac{1}{2}\ln ...
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24 views

What is the functional equivalent of this matrix multiplication, if any?

I have a simple short Mathematica program that does the following: First we define a lower triangular matrix $A$ with the defintion: If $k$ divides $n$: $A(n,k) = k^{(1/2 - x)}$ else $A(n,k)=0$ ...
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3answers
228 views

Zeta function summation

Looking at the question here Surprising identities / equations shows an interesting identity, $$\large (\zeta (2)-1)-(\zeta (3)-1)+(\zeta(4)-1)-\cdots=\frac 12$$ How can we prove that result, or even ...
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3answers
58 views

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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35 views

Divergence Dedekind zeta function

Let $K$ be a number field, $\mathcal O_K$ be its ring of integers, T a positive integer and $N$ the norm function. Give an upper bound (in T) for $$\sum_{I\leq \mathcal O_K: N(I)\leq T} ...
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1answer
52 views

What function satisfies the following equation?

$$f(x)e^{-x}\Gamma(x/\pi)=f(\pi/2-x)e^{x-\pi/2}\Gamma(1/2-x/\pi)$$ I think it should be similar to Zeta function, but what is it exactly?