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

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22
votes
2answers
410 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.
3
votes
1answer
180 views

Dirichlet series and Riemann zeta function

Im trying to show, for $\Re(s)>1$, that $\displaystyle\sum_{n=0}^{\infty} \frac{d(n^2)}{n^s} = \frac{\zeta^3(s)}{\zeta(2s)}$, where $d(n)= |\{k \mid k|n \}|$, number of positive integers that ...
0
votes
0answers
100 views

Any underlying reason why these equations look similar? [on hold]

Questions Is there any way to go from either of these equations to the other? Or is there any more fundamental reason for their similarities? $$ \frac{1}{\zeta(s)} = \sum_{r=1}^\infty \frac{\mu(r)}{...
68
votes
10answers
12k views

Nice proofs of $\zeta(4) = \pi^4/90$?

I know some nice ways to prove that $\zeta(2) = \sum_{n=1}^{\infty} \frac{1}{n^2} = \pi^2/6$. For example, see Robin Chapman's list or the answers to the question "Different methods to compute $\sum_{...
1
vote
2answers
66 views

Inequality involving sum of logarithms and hidden zeta-function

I would like to prove the following estimation: if $n \ge 2$ is a natural number, then $$\sum_{k=2}^n \frac{\log^2 k}{k^2} <2 - \frac{\log^2 n}{n}.$$ I have noticed that LHS is indeed bounded by ...
2
votes
1answer
33 views

Laplace transform of bell-shaped functions

A real smooth function $\varphi$ is said bell shaped iff as the Gaussian : $\varphi''$ is positive on $(-\infty,a) \cup (b,+\infty)$ and negative on $(a,b)$. I'm interested in the bilateral Laplace ...
7
votes
1answer
196 views

An Inequality Involving The Riemann Zeta Function

I'm having trouble proving the following inequality for $2<r<3$: $$(1+2^{-r})\frac{(3^r+1)^2}{3^{2r}+1}>\frac{\zeta(r)}{\zeta(2r)}.$$ I can easily plot the graph, and the inequality clearly ...
-2
votes
0answers
65 views

It is possible to use the Zeta Function as primality test? [closed]

It is possible to use the Zeta Function as primality test? $$\displaystyle\sum_{n=1}^\infty\dfrac1{n^s} = 1+\frac{1}{1^s}+\frac{1}{2^s}+\frac{1}{3^s}+ ... $$ Where can I find the non-trivial zeros ...
4
votes
1answer
73 views

(Non-)Canonicity of using zeta function to assign values to divergent series

This article http://blogs.scientificamerican.com/roots-of-unity/does-123-really-equal-112/ got me thinking about the "identity" $$1 + 2 + 3 + \cdots = -1/12,$$ and I wanted to convince myself there ...
3
votes
1answer
54 views

Prime Number Theorem and the Riemann Zeta Function

Let $$\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}$$ be the Riemann zeta function. The fact that we can analytically extend this to all of $\mathbb{C}$ and can find a zero free region to the left of the ...
0
votes
1answer
32 views

Prove that this limit is the logarithmic derivative of the Riemann zeta function.

Prove the following limit: $$-\frac{\zeta '(s)}{\zeta (s)}=\lim_{c\to 1} \, \left(\frac{\zeta (c) \zeta (s)}{\zeta (c+s-1)}-\zeta (c)\right)$$ As a starting point I tried to enter this series ...
3
votes
4answers
112 views

Evaluating series of zeta values like $\sum_{k=1}^{\infty} \frac{\zeta(2k)}{k16^{k}}=\ln(\pi)-\frac{3}{2}\ln(2) $

Somehow I derived these values a few years ago but I forgot how. It cannot be very hard (certainly doesn't require "advanced" knowledge) but I just don't know where to start. Here are the sums: $$ \...
1
vote
0answers
49 views

Riemann's zeta function in a definite integral

I am trying to evaluate the value of the definite integral $ I= \int_0^1((1-\delta)\log[p]-\delta\log[1-p])^4dp$ Using Binomial expansion I get $I=(1-\delta)^4\int_0^1(\log[p])^4dp-4(1-\delta)^3\...
7
votes
2answers
473 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 $$\sum_{n=1}^\...
13
votes
5answers
511 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).
0
votes
1answer
45 views

What is the equivalent statement of GRH in term of Redheffer Matrix or Farey Sequences?

We all know that Riemann Hypothesis (RH) has many equivalent statements. There is one statement which expresses RH in term of Redheffer matrix, there is another equivalent statement of RH which ...
0
votes
1answer
74 views

How was the integral for Zeta Function created

How was the zeta function integrated from $$\zeta(s) = \sum_{n=1}^{\infty}\frac{1}{n^{s}}$$ To $$\zeta(s) = \frac{1}{\Gamma (s)}\int_{0}^{\infty}\frac{x^{s-1}}{e^{x}-1}dx$$ I've tried googling ...
1
vote
0answers
34 views

Is there any product formula for local zeta function?

Suppose that $V$ is a non-singular $n$-dimensional projective algebraic variety over the field $\mathbb{F}_q$ with $q$ elements. The local zeta function $Z(V, s)$ of $V$ (sometimes called the ...
0
votes
1answer
25 views

Is the functional equation for the inverse of Riemann zeta function valid for any point in the critical strip?

It is known that in the critical strip $s\in \{0<\mathrm{Re}(s)<1\}$,Riemann zeta function satisfies the following functional equation: $$\zeta(s)=\chi(s)\zeta(1-s),\tag{1}$$ $$\chi(s)=\frac{\...
1
vote
1answer
67 views

Riemann Zeta Function, Stirling's Numbers, and Infinite Series

A while back I was able to prove the following identity, $$\sum_{k=1}^{\infty}\frac{\Gamma(k+r)}{\Gamma(k)(k+r)^s}=\sum_{k=0}^{r}s(r+1,r+1-k)\zeta(s-r+k)$$ where $s(k,n)$ are the Sterling numbers of ...
1
vote
1answer
59 views

Is $\zeta(2n+1)\notin (2\pi)^{2n+1}\mathbb{Q}$ already known?

Is it already shown or at least conjectured that $$\zeta(2n+1)\notin (2\pi)^{2n+1}\mathbb{Q}?$$ You have any names and years who proved or conjectured it?
18
votes
1answer
498 views

Proof of $\sum_{n=1}^\infty \frac{1}{n^4 \binom{2n}{n}}=\frac{17\pi^4}{3240}$

Recently, I was able to prove that $$\sum_{n=1}^\infty \frac{1}{n \binom{2n}{n}}= \frac{\pi}{3\sqrt{3}}$$ $$\sum_{n=1}^\infty \frac{1}{n^2 \binom{2n}{n}}= \frac{\pi^2}{18}$$ But does anybody know ...
3
votes
2answers
46 views

What is already known for $\zeta(n)$, $n\in 2\mathbb{N}+1$

Apéry showed $\zeta(3)\notin\mathbb{Q}$. What is also known or conjectured for $\zeta(n)$ with n odd? Is for example something known for $\zeta(5)$? Is there a theorem that says 'at least one of $\...
3
votes
0answers
108 views

counterexample to RH; how big would it have to be?

If the Riemann hypothesis is false, then there has to be a first counterexample for $\zeta(z)=0$ in the critical strip with $\Re(z) \ne \frac{1}{2}$. For such a counterexample, how large would $T=|\...
2
votes
0answers
50 views

Why are we interested in such things as $\zeta(3)\notin\mathbb{Q}$? [closed]

In 17xx Euler gived a formula for the real numbers $\zeta(2n),~n\ge 1,$ which showed the irrationality of $\zeta(2n)$. In 1975 Apéry showed $\zeta(3)\notin\mathbb{Q}$. Why are we interested in such ...
2
votes
1answer
71 views

Question on a proof of $\zeta(3)\notin\mathbb{Q}$

I have a question on this article proving $\zeta(3)\notin\mathbb{Q}.$ by using modular forms. This is theorem 1 at page 275 (page 5 in the pdf). Most things in the proof are clear but I don't get the ...
0
votes
1answer
56 views

When did Euler find his formula for $\zeta(2n)$

Does anybody know when Euler found his famous formula $$\zeta(2n)=\frac{(-1)^{n-1}(2\pi)^{2n}B_{2n}}{2(2n)!}?$$
1
vote
0answers
53 views

Xi Function on Critical Strip - Mellin Transform

Story I'm trying to prove the following identity $$\int_0^\infty \frac{\Xi(t)}{t^2 + \frac{1}{4}} \cos(xt) dt = \frac{1}{2} \pi (e^{\frac{1}{2}x} - 2e^{-\frac{1}{2}x} \psi(e^{-2x}))$$ where $$\psi(...
0
votes
1answer
59 views

Contour Integration, Riemann Zeta (-n)

I was reading Riemann's Zeta Function by H. Edwards, and could not understand the equation on the page 12. \begin{align*} \zeta(-n) &= \frac{\prod(n)}{2\pi i}\int_{+\infty}^{+\infty} \frac{(-x)^{-...
2
votes
0answers
36 views

Convergence of sequence with $\zeta$ function

Last time I heard interesting question. Unfortunately I do not have idea how to solve it, so I decided to give it here. Let us define sequence $a_n=(\underbrace{\zeta\circ...\circ \zeta}_{n})(\pi)$ ...
0
votes
2answers
92 views

Riemann zeta function functional equation proof explanation

In Riemann zeta function functional equation proof I arrived to a following equation $$\frac{\Gamma\left(\frac{s}2\right) \zeta(s)}{\pi^{\frac{s}2}}=\sum_{n=1}^\infty \int_0^\infty x^{\frac{s}2-1}e^{-...
1
vote
0answers
63 views

Is this Dirichlet series generating function of the von Mangoldt function matrix correct?

Let $\mu(n)$ be the Möbius function and let $a(n)$ be the Dirichlet inverse of the Euler totient function: $$a(n) = \sum\limits_{d|n} d \cdot \mu(d)$$ Let the matrix $T$ be defined as: $$T(n,k)=a(...
1
vote
1answer
52 views

Deriving $ \frac{1}{n^s} = \frac{1}{\Gamma(s)} \int_0^{\infty}t^{s-1} e^{-nt} \, dt $ Backwards?

Is it possible to start with $\dfrac{1}{n^s}$ and then, without knowing the Gamma function in advance, naturally (with reasons!) derive that $$ \frac{1}{n^s} = \frac{1}{\Gamma(s)} \int_0^{\infty}t^{s-...
3
votes
1answer
165 views

Computing Zeta(2)

It is well known that $$\int_0^1\frac{\log{x}}{1 - x}\,\mathrm{d}x = -\frac{\pi^2}{6} $$ This is generally proved by expanding the geometric series and then using $\zeta(2) = \pi^2/6$. My question ...
1
vote
1answer
48 views

Searching Riemann zeta zeros in nuclear data files.

Is this match between some of the truncated Riemann zeta zeros and numbers in nuclear calculation data only a coincidence? I calculated these numbers from the Riemann zeta zeros and looked them up in ...
7
votes
1answer
246 views

Connection between integral expression and the factorial of infinity

Does the fact that $$\int_{-\infty}^{\infty}\exp\left(-\frac{1}{2}x^2\right)\mathrm{d}x=\sqrt{2\pi}$$ Have something to do with the fact that the regularized factorial of infinity is also $\sqrt{2\...
3
votes
1answer
130 views

Series involving the Riemann zeta function

Consider the series: $$\sum_{n=1}^{\infty}\frac{\zeta(2n+1)}{n(2n+1)}$$ We can easily prove that it's a convergent series. My question, is there a way to express this series in terms of zeta ...
5
votes
0answers
274 views

An interesting identity involving powers of $\pi$ and values of $\eta(s)$

It happens that, for any $m\geq 1$, $$\sum_{n=0}^{+\infty}\frac{(-1)^n}{(2n+1)^{2m+1}}=\frac{E_{2m}}{2\cdot(2m)!}\left(\frac{\pi}{2}\right)^{2m+1}\tag{1}$$ where $E_{2m}$ is an integer number. My ...
2
votes
1answer
91 views

Ramanujan's divergent series

I tried to prove this sum by myself, but I couldn't. $1 + 4 + 9 + 16 + ... = 0$ First, I know this sums are a bit problematic, as we can't just $'='$ an infinite sum, but I would like to see the ...
0
votes
0answers
51 views

What's about the derivative of the Riemann zeta function?

The derivative of the Riemann Zeta function is $$\zeta'(s)=-\sum_{n=2}^\infty\frac{\log n}{n^s}$$ for $\Re s>1$. Question. Can you refers us in a short post, from a divulgative viewpoint (but ...
9
votes
1answer
191 views

Riemann zeta function and the volume of the unit $n$-ball

The volume of a unit $n$-dimensional ball (in Euclidean space) is $$V_n = \frac{\pi^{n/2}}{\frac{n}{2}\Gamma(\frac{n}{2})}$$ The completed Riemann zeta function, or Riemann xi function, is $$\xi(s) ...
15
votes
1answer
234 views

Connection between the area of a n-sphere and the Riemann zeta function?

The Riemann Xi-Function is defined as $$ \xi(s) = \tfrac{1}{2} s(s-1) \pi^{-s/2} \Gamma\left(\tfrac{1}{2} s\right) \zeta(s) $$ and it satisfies the reflection formula $$ \xi(s) = \xi(1-s). $$ But the ...
8
votes
3answers
392 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,
5
votes
3answers
105 views

Prove $\int_0^{\infty} \frac{x^2}{\cosh^2 (x^2)} dx=\frac{\sqrt{2}-2}{4} \sqrt{\pi}~ \zeta \left( \frac{1}{2} \right)$

Wolfram Alpha evaluates this integral numerically as $$\int_0^{\infty} \frac{x^2}{\cosh^2 (x^2)} dx=0.379064 \dots$$ Its value is apparently $$\frac{\sqrt{2}-2}{4} \sqrt{\pi}~ \zeta \left( \frac{1}...
2
votes
1answer
76 views

On a second-order differential inequality involving the Dirichlet eta function

After that I've tried understand the problem 6416 [1983, 60] A Second-Order Differential Inequality proposed by Sandford S. Miller in the American Mathematical Monthly (myself proposal is ...
58
votes
1answer
2k views

Evaluating the log gamma integral $\int_{0}^{z} \log \Gamma (x) \, \mathrm dx$ in terms of the Hurwitz zeta function

One way to evaluate $ \displaystyle\int_{0}^{z} \log \Gamma(x) \, \mathrm dx $ is in terms of the Barnes G-function. $$ \int_{0}^{z} \log \Gamma(x) \, \mathrm dx = \frac{z}{2} \log (2 \pi) + \...
11
votes
2answers
706 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 ...
0
votes
1answer
36 views

Why can't an argument for the Riemann Zeta function be 1? What happens if we take Re(s)=1? [duplicate]

If $s=1$, then the series equals to $\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+...\to \infty$ This certainly does seem to be a convergent series. Why doesn't it have a limit?
2
votes
0answers
26 views

Analyitc Continuation of Partial $\zeta$ function

Let $A\subset \mathbb{N}$. Consider the partial $\zeta$ function $$\sum_{a\in A} \frac{1}{a^s}$$ We know this converges for Re$(s)>1$. Under what conditions of $A$, can this be analyically ...
3
votes
1answer
143 views

infinity series of Riemann zeta function at odd integers

Properties of Riemann zeta function at odd and even integers diverge dramatically, which can be proved by many evidences. I once found an infinity series in wikipedia, it reads $$ \sum_{n=1}^{\infty}...