2
votes
1answer
69 views

Question on the Prime Number Theorem (the Tchebychev Function) [duplicate]

This has been giving me nothing but a headache: Let the Tchebychev Function, $\psi (x)$ be defined: $$\psi (x) = \sum_{p^m \le x}\log p \space \space \space , \space \space \space p \in \mathbb P$$ ...
4
votes
2answers
143 views

Zeta function for negative integers

I already proved that $\zeta(z)=\frac{1}{\Gamma(z)}\int_0^\infty\frac{t^{z-1}}{e^t-1}dt=\frac{\Gamma(z-1)}{2\pi i}\int_{-\infty}^0\frac{t^{z-1}}{e^{-t}-1}dt$ Now the Benoulli numbers are defined by ...
3
votes
0answers
46 views

How can I integrate this zeta function expression?

Can you integrate this function: $$f(k)=\exp\left(-\Re\left(\sum\limits_{n=1}^{n=scale} \frac{1}{n} \zeta(1/2+i \cdot k)\sum\limits_{d|n} \frac{\mu(d)}{d^{(1/2+i \cdot k-1)}}\right)\right)$$ with ...
6
votes
2answers
88 views

Ways to calculate $\int_0^1 \frac{-\log x}{1+x}\ \mathrm dx$

I came across the integral $$ \int_0^1 \frac{-\log x}{1+x}\ \mathrm dx = \frac{\pi^2}{12}, $$ which can be calculated as $\frac 1 2 \zeta(2)$ using analytic number theory. I'm interested if this ...
13
votes
2answers
245 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 $$\int_0^\frac{1}{2}x\cot(\pi x)\,dx=\frac{\log(2)}{2\pi}$$ ...
2
votes
2answers
37 views

Limit $(t-1)\zeta(t)$ as $t\rightarrow 1^+$

Show that $\lim_{t\rightarrow 1^+}(t-1)\zeta(t)=1$. For $t>1$, we can use the definition $\zeta(t)=\sum_{n=1}^\infty \dfrac{1}{n^t}$, so it is approximately $\int_1^\infty \dfrac{1}{x^t}dx$. ...
2
votes
1answer
56 views

Is this formula for $\zeta(2n+1)$ correct or am I making a mistake somewhere?

I am calculating $\zeta(3)$ from this formula: $$\zeta(2n+1)=\frac{1}{(2n)!}\int_0^{\infty} \frac{t^{2n}}{e^t -1}dt$$ From Grapher.app, I get $\int_0^{\infty} \frac{x^{2}}{e^x -1}dx = .4318$ ...
11
votes
0answers
411 views

The log gamma integral $\int_{0}^{z} \log \Gamma (x) \ \mathrm dx$

One way to evaluate $ \displaystyle\int_{0}^{z} \log \Gamma(x) \ \mathrm dx $ is in terms of the Barnes G function. $$ \int_{0}^{z} \ln \Gamma(x) \ \mathrm dx = \frac{z}{2} \log (2 \pi) + ...
55
votes
3answers
1k views

A math contest problem $\int_0^1\ln\left(1+\frac{\ln^2x}{4\,\pi^2}\right)\frac{\ln(1-x)}x \ \mathrm dx$

A friend of mine sent me a math contest problem that I am not able to solve (he does not know a solution either). So, I thought I might ask you for help. Prove: ...
0
votes
0answers
65 views

Zeroes of $s+\sum\limits_{n=2}^\infty \frac{(-1)^{n+1}}{n^s\ln n} $?

Where are the solutions of the equations $$s+\sum\limits_{n=2}^\infty \dfrac{1}{n^s\ln n}=0\quad \text{and}\quad s+\sum\limits_{n=2}^\infty \dfrac{(-1)^{n+1}}{n^s\ln n}=0 ?$$ Since the ...
16
votes
3answers
304 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 ...
1
vote
0answers
71 views

Convergence of an integral involving the fractional part function

It's not too hard to show that $\displaystyle \int_{\textbf{1}}^{\infty} \frac{\text{frac}(x)}{x^{s+1}} \ dx = \frac{1}{s-1} - \frac{\zeta(s)}{s}$ for $s >1$. And if you take the limit of both ...
17
votes
1answer
289 views

References to integrals of the form $\int_{0}^{1} \left( \frac{1}{\log x}+\frac{1}{1-x} \right)^{m} \, dx$

While extending my calculation techniques, with aid of Mathematica, I found that \begin{align*} \int_{0}^{1}\left( \frac{1}{\log x} + \frac{1}{1-x} \right)^{3} \, dx &= -6 \zeta '(-1) ...
3
votes
1answer
125 views

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

I'm trying to find to value of: $$ J=\int_{0}^{\infty}\frac{x^3}{e^x-1}\ln(x)\,dx $$ Here's what I've done: $$ ...
3
votes
2answers
144 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.
11
votes
1answer
298 views

Closed form for $\sum_{n=2}^\infty \frac{1}{n^2\log n}$

I had attempted to evaluate $$\int_2^\infty (\zeta(x)-1)\, dx \approx 0.605521788882$$ Upon writing out the zeta function as a sum, I got $$\int_2^\infty ...
10
votes
2answers
557 views

How to show that the Laurent series of the Riemann Zeta function has $\gamma$ as its constant term?

I mean the Laurent series at $s=1$. I want to do it by proving $\displaystyle \int_0^\infty \frac{2t}{(t^2+1)(e^{\pi t}+1)} dt = \ln 2 - \gamma$, based on the integral formula given in Wikipedia. ...
12
votes
1answer
400 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 = ...
0
votes
0answers
139 views

Determining well-definedness for functions

How does one determine well-definedness in analytical continuation for $\Gamma(s)\zeta(s)$ function? Firstly: $$\Gamma(s)\zeta(s) = \int_0^\infty dt \frac{t^{s-1}}{e^t - 1},\quad Re(s) > 1$$ ...
10
votes
2answers
472 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 ...