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

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1answer
60 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?
7
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1answer
80 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 ...
4
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1answer
71 views

Strange sum involving $\zeta (3)$

$$\sum _{l=0}^{\infty } \sum _{k=0}^{\infty } \frac{(-1)^{k+l}}{2 (l+1)^3 \left((k+1)^3+(l+1)^3\right)}=\frac{9 \zeta (3)^2}{32}$$ I would like to prove the proposed equality, but I do not know where ...
2
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2answers
23 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
89 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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1answer
20 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 ...
1
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1answer
99 views

Can the Riemann Zeta derivative be expressed in terms of Riemann Zeta?

From this question: http://mathoverflow.net/questions/134600/zetax-in-terms-of-zetax-zeta1-x-gamma-psi it seems that Zeta can be expressed through its derivative: $$\zeta(1-x) = ...
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0answers
70 views

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 ...
8
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1answer
124 views

Infinite Series $\sum_{n=-\infty}^{\infty}\frac{1}{(x+n\pi)^m}$

How can we find a closed form for the following infinite series for any $m\in\mathbb N$? $$\sum_{n=-\infty}^{\infty}\frac{1}{(x+n\pi)^m}$$
2
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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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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 ...
0
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0answers
35 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 ...
1
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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 = ...
7
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1answer
72 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 ...
12
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2answers
261 views

Infinite Series $\sum_{n=1}^\infty\frac{\zeta(3n)}{2^{3n}}$

I want to evaluate $$\sum_{n=1}^\infty\frac{\zeta(3n)}{2^{3n}}$$ let $f(z)=\sum_{n=1}^\infty\frac{1}{2^{3n}}z^{3n}$, then $$\sum_{n=1}^\infty ...
1
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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 ...
2
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1answer
109 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) = ...
1
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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 ...
2
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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 $$ ...
-1
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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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6answers
1k views

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

‎If $f\left(z \right)=\sum_{n=2}^{\infty}a_{n}z^n$ and $\sum_{n=2}^{\infty}|a_n|$ converges then‎, $$\sum_{n=1}^{\infty}f\left(\frac{1}{n}\right)=\sum_{n=2}^{\infty}a_n\zeta\left(n\right)‎.$$ ‎Since ...
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1answer
377 views

Is this integral $\int_0^1\left(\left\{\frac1x\right\}-\frac12\right)\frac{\log(x)}{x}dx$ equal to zero?

My initial question was to find if this integral $$ \int_0^1 \left(\left\{\frac 1x\right\}-\frac12\right)\frac{\log(x)}{x}dx$$ is convergent or divergent. ($\left\{\frac 1x\right\}$ is the fractional ...
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0answers
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 ...
1
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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 $t\in \mathbb R$,$x\geq |t|/\pi\implies \zeta(s)=\displaystyle \sum_{n\leq x} \frac{1}{n^s}+\frac{x^{1-s}}{s-1}+O(x^{-\sigma})$ ...
15
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1answer
292 views

Infinite Series $\sum_{n=1}^\infty\frac{H_{2n+1}}{n^2}$

How can I prove that $$\sum_{n=1}^\infty\frac{H_{2n+1}}{n^2}=\frac{11}{4}\zeta(3)+\zeta(2)+4\log(2)-4$$ I think this post can help me, but I'm not sure.
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0answers
207 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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0answers
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 $\zeta(n)=-\frac {B_{n+1}} {n+1}$. ...
4
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0answers
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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0answers
53 views

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, ...
11
votes
2answers
391 views

Using the rules that prove the sum of all natural numbers is $-\frac{1}{12}$, how can you prove that the harmonic series diverges?

I think I understand intuitively how we can assign a value to the sum of all natural numbers. But of all the proofs that I've seen that show why $\zeta(-1) = -\frac{1}{12}$, none of them use their own ...
3
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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} ...
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 ...
11
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0answers
414 views

I can Euler-sum $\sqrt{-\ln(1)}-\sqrt{-\ln(2)}+\sqrt{-\ln(3)}-\cdots$. But how can I do $\sqrt{-\ln(1)}+\sqrt{-\ln(2)}+\sqrt{-\ln(3))}+\cdots$?

This is also related to an older thread in MSE ("what is the half derivative of zeta at zero?") . One of the possible steps in the problem of that thread was to evaluate the series ...
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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 ...
0
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1answer
60 views

How to introduce an integer function into $\zeta$ function instead of $n$

I have a problem that I am struggling with since long and probably it is simple but I can not get through. So your help would be very welcome. Known that Riemann $\zeta$ function is defined as sum ...
4
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0answers
63 views

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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1answer
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 ...
3
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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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0answers
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$$\gamma_{1}=14.13...$$ ...
2
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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 ...
0
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1answer
49 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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1answer
502 views

Ramanujan Summation not consistent with Riemann's Zeta function?

Wikipedia states that Ramanujan sums and the Riemann Zeta function have the same values for even $k$: $$1 + 2^{2k} + 3^{2k} + \cdots = 0\ (\Re)$$ However, I don't understand how this can be true, ...
5
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1answer
62 views

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 ...
0
votes
0answers
53 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 $-\frac 1 {12}$. ...
0
votes
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?
1
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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 $\zeta(s) = a$ ...