# Questions tagged [periods]

A period is a number that can be expressed as an integral of an algebraic function over an algebraic domain.

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### Automorphisms of the ring of periods

The set of periods $\mathcal{P}$ introduced by Kontsevich and Zagier forms a ring, see for example https://en.m.wikipedia.org/wiki/Period_(algebraic_geometry). Moreover J. Wan introduced in 2011 in ...
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### Showing that $\int_0^\pi\frac{x\ln(1-\sin x)}{\sin x}dx=3\int_0^\frac{\pi}{2}\frac{x\ln(1-\sin x)}{\sin x}dx$

Prove, without evaluating the integrals that: $$\int_0^\pi\frac{x\ln(1-\sin x)}{\sin x}dx=3\int_0^\frac{\pi}{2}\frac{x\ln(1-\sin x)}{\sin x}dx$$ Originally I posted this here on MSE, however it's ...
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### 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 ...
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### Is special value of Epstein zeta function in 3 variables a period?

Kontsevich-Zagier's article "Periods" contains the following question Is $\displaystyle \sum_{x,y,z \in \mathbb{Z}}' \frac{1}{(x^2+y^2+z^2)^2}$ an extended period? ($\sum'$ means we do not sum ...
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### What are reasons to believe that e is not a period?

In their 2001 paper defining periods, Kontsevich and Zagier (pdf) without further comment state that $e$ is conjecturally not a period while many other numbers showing up naturally (conjecturally) are....
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### What is the probability that a randomly chosen number from set of c.e.number is period(number)?

What is the probability that a randomly chosen number from the set of c.e.numbers is period(number)? What is the probability that a randomly chosen number from the set of computable numbers is ...
The motivation for this question is to understand a recent theorem of Francis Brown which implies that all periods of mixed Tate motives over $\mathbb{Z}$ lie in $\mathcal{Z}[\frac{1}{2\pi i}]$, where ...