7
votes
3answers
161 views

Hard Definite integral involving the Zeta function

Prove that: $$\displaystyle \int_{0}^{1}\frac{1-x}{1-x^{6}}{\ln^4{x}} \ {dx} = \frac{16{{\pi}^{5}}}{243\sqrt[]{{3}}}+\frac{605\zeta(5)}{54} $$ I was able to simplify it a bit by substituting ${y = ...
2
votes
0answers
45 views

Intuitive explanation for $\zeta (2)=\frac{\pi^2}{6}$ [duplicate]

Using $f(x)=x^2$ Fourier' series, the proof for $\zeta (2)=\frac{\pi^2}{6}$ is pretty straight forward. I'm wondering if there is a more intuitive explanation for the equality, one that a layman could ...
4
votes
1answer
71 views

Limit of differences of truncated series and integrals give Euler-gamma, zeta and logs. Why?

In the MSE-question in a comment to an naswer Michael Hardy brought up the following well known limit- expression for the Euler-gamma $$ \lim_{n \to \infty} \left(\sum_{k=1}^n \frac 1k\right) - ...
7
votes
0answers
64 views

A cotangent series related to the zeta function

$$\sin x = x\prod_{n=1}^\infty \left[1-\frac{x^2}{n^2\pi^2}\right]$$ If you apply $\log$ to both sides and derivate: $$\cot x = \frac{1}{x} - \sum_{n=1}^\infty \left[\frac{2x}{n^2\pi^2} ...
0
votes
0answers
19 views

Difference between Eulers product and Zeta Function at a finite values

So a very important formula proven by Euler is that is equal to of course these formulas give you the same value when they reach infinity, but my question is that say s=1. What would be the ...
0
votes
0answers
41 views

Why there's no articles about the eta function convergence?

I've been searching about a proof that the eta function converges for $\mbox{Re}(z)>0$ but the ONLY page I've found that claims to prove it was in this question: ...
0
votes
0answers
29 views

What is the sign of the generalized Stieltjes constants $\gamma_{k}(a)$?

Recall that the Stieltjes constants $\gamma_{k}$ appear as the coefficients in the regular part of the Laurent expansion of the Riemann zeta function about $s = 1$: $$ \begin{align} \zeta(s) = ...
0
votes
0answers
62 views

What does this complex contour integral represent?

How would one evaluate the following complex contour integral in "Integral and Series Representations of Riemann’s Zeta function, Dirichelet’s Eta Function and a Medley of Related Results." The ...
6
votes
2answers
146 views

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

We have $$ \sum_{j=1}^{\infty}\sum_{i=1}^{\infty} \displaystyle \frac{(i-1)! (j-1)!}{(i+j)!} H_{i+j} =\displaystyle 3 \: \zeta(3) $$ where $\displaystyle H_{n}:=\sum_{1}^{n} \frac{1}{k}$ are ...
6
votes
2answers
154 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 $$ \displaystyle \sum_{n=1}^{+ \infty} \left(H_{n}-\ln n-\gamma -\frac{1}{2n}\right)=\frac{1}{2} \left(1-\ln (2\pi)+\gamma\right) \quad (*) $$ where $H_{n}:=\sum_{1}^{n} ...
5
votes
0answers
79 views

Generating function of the squared Riemann zeta function

It's a well known fact that $$\sum_{k=2}^{\infty} \zeta(k) x^k=-x \psi(1-x)-x\gamma \space (|x|<1) $$ but I didn't meet yet a version for squared Riemann zeta function $$\sum_{k=2}^{\infty} ...
13
votes
2answers
277 views

Closed form of $\int_0^\frac{1}{2}x^n\cot(\pi x)\,dx$

What is the closed form of the following integral $$\int_0^\frac{1}{2}x^n\cot(\pi x)\,dx,n\in\mathbb{N}$$ By Mathematica I saw that $$\int_0^\frac{1}{2}x\cot(\pi x)\,dx=\frac{\log(2)}{2\pi}$$ ...
6
votes
3answers
227 views

Question about Euler's approach to find $\sum_{n=1}^\infty\frac1{n^2}=\frac{\pi^2}6$

For a freshman calculus project, I used Euler's approach to find $\sum_{n=1}^\infty\frac1{n^2}=\frac{\pi^2}6$, and noted from Wikipedia's explanation that the infinite product representation of ...
17
votes
3answers
353 views

Find the value of $\int_{0}^{\infty}\frac{x^3}{e^x-1}\ln(e^x - 1)\,dx$

I'm trying to figure out how to evaluate the following: $$ J=\int_{0}^{\infty}\frac{x^3}{e^x-1}\ln(e^x - 1)\,dx $$ I'm tried considering $I(s) = \int_{0}^{\infty}\frac{x^3}{(e^x-1)^s}\,dx\implies ...
4
votes
1answer
160 views

question about Riemann zeta $\zeta (0)$ [duplicate]

i know that $$\zeta (m)=\sum_{n=1}^\infty n^{-m}$$ so $$\zeta (0)=\sum_{n=1}^\infty n^0=1+1+1+1+1+1+\cdots=\infty $$ but actually $$\zeta (0)=-0.5$$ where is the wrong please help thanks ...
1
vote
2answers
124 views

Proof that the zeta function converges for Re(s)>1

It would be absolutely fantastic if anybody could give me some guidance on the question above. For me (please correct me if I'm wrong), this question boils down to proving that ...
7
votes
2answers
343 views

A tough series: $\sum_{k=1}^{\infty}\frac{\zeta(2k)}{(k+1)(2k+1)}$, need help

I was doing a integral which ends up with a tough series part: $$\sum_{k=1}^{\infty}\frac{\zeta(2k)}{(k+1)(2k+1)}$$ Mathematica says $$\frac12$$ Which agrees with the anwer...Anyone know how to ...
3
votes
2answers
149 views

Another improper integral

Show that : $$\int_0^1\frac{(\sin ^{-1}x)^2}{x}\text{d}x=\frac{\pi ^2\ln 2}{4}-\frac78\zeta(3)$$ This integral is in "irresistible integrals" on page 122. I can't prove this one.
2
votes
0answers
194 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( ...
9
votes
1answer
254 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
366 views

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 ...
3
votes
2answers
336 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 ...
3
votes
2answers
250 views

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 ) ...
21
votes
3answers
1k views

Computing $\zeta(6)=\sum\limits_{k=1}^\infty \frac1{k^6}$ with Fourier series.

Let $ f$ be a function such that $ f\in C_{2\pi}^{0}(\mathbb{R},\mathbb{R}) $ (f is $2\pi$-periodic) such that $ \forall x \in [0,\pi]$: $$f(x)=x(\pi-x)$$ Computing the Fourier series of $f$ and ...
12
votes
1answer
438 views

challenging integral involving $\zeta(5)$

I ran across a curious integral that seems to be rather tough that some on the site may enjoy. Show that $$\displaystyle \int_{0}^{1}\frac{\sqrt{1-x^{2}}}{1-x^{2}\sin^{2}(x)}dx = ...
10
votes
2answers
497 views

Integrating $\frac{x^k }{1+\cosh(x)}$

In the course of solving a certain problem, I've had to evaluate integrals of the form: $$\int_0^\infty \frac{x^k}{1+\cosh(x)} \mathrm{d}x $$ for several values of k. I've noticed that that, for k a ...
41
votes
7answers
6k 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 ...