All Questions
Tagged with integration riemann-zeta-function
9 questions
22
votes
2
answers
1k
views
A closed form for an integral expressed as a finite series of $\zeta(2k+1)$, $\pi^m$ and a rational?
In this paper the following beautiful integral expression for $\zeta(3)$ is derived:
$$\zeta(3)=\frac{1}{7}\,\int_0^{\pi} x\,(\pi-x)\csc(x)\, dx$$
In a comment at the end of this question, I ...
- 4,169
29
votes
2
answers
2k
views
Is there a closed form for $\int_0^\infty\frac{\tanh^3(x)}{x^2}dx$?
For $n\geqslant m>1$, the integral $$I_{n,m}:=\int\limits_0^\infty\dfrac{\tanh^n(x)}{x^m}dx$$ converges. If $m$ and $n$ are both even or both odd, we can use the residue theorem to easily evaluate ...
- 13.4k
27
votes
2
answers
2k
views
Are these two new ways of representing odd zeta values as integrals known?
This is inspired by the same beautiful integral expression for $\zeta(3)$ as this question, but goes in a slightly different direction. Writing the original integral in the form $$\int_0^1\frac{x(1-x)}...
- 13.4k
13
votes
3
answers
823
views
Is there a closed form of $\int_0^\frac12\dfrac{\text{arcsinh}^nx}{x^m}dx$?
For naturals $n\ge m$, define
$$I(n,m):=\int_0^\frac12\dfrac{\text{arcsinh}^nx}{x^m}dx$$
with $\text{arcsinh}\ x=\ln(x+\sqrt{1+x^2} )$, so $\text{arcsinh} \frac12=\ln \frac{\sqrt{5}+1}2 $.
Is it ...
- 13.4k
13
votes
2
answers
743
views
How to prove that $\int _0^\infty\frac{\text{arcsinh}^nx}{x^m}dx$ is a rational combination of zeta values?
For $n\ge m\ge 2$, define $$I(n,m):= \int _0^\infty\dfrac{\text{arcsinh}^nx}{x^m}dx$$ Computer algebra systems say that the indefinite integral can be expressed in terms of polylog functions (of ...
- 13.4k
6
votes
0
answers
292
views
A kind of reflection formula for the logarithmic derivative of the zeta function
So I was messing around with Bernoulli numbers and values of $\zeta'$ at integers $-$ and suddenly I came about a non trivial identity which can be written in terms of the logarithmic derivative of ...
- 13.4k
3
votes
0
answers
231
views
Could the patterns in the roots of the Riemann-Liouville differintegral for $(s-1)\,\zeta(s)$ be explained?
The well-known integral expression for the entire function:
$$(s-1)\,\zeta(s) = \frac{-i\,\pi}{2}\int_{1/2-i\infty}^{1/2+i\infty} \frac{\csc(\pi\,u)^2}{u^{s-1}} \, du \qquad s \in \mathbb{C} \tag{0}$$
...
- 4,169
3
votes
1
answer
344
views
Another integral that has a closed form involving finite series of $\zeta(2k+1)$'s. Could it be reflexive?
In the context of a series of questions here, here and here, about closed form expressions involving finite series of $\zeta(2k+1)$'s for certain integrals, I would like to raise another one:
$$f(n):=...
- 4,169
2
votes
0
answers
361
views
Definite integral probably equal to zeta with known (but unusable) closed form for the indefinite integral
Related to this and
this questions.
Basically got definite integral that experimentally equals
$\zeta(s)$ both numerically and symbolically.
Closed form for the indefinite integral is known, but I ...
- 25.4k