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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1answer
545 views

Quaternionic and octonionic analogues of the Basel problem

I asked this question in MSE around 3 months ago but I have received no answer yet, so following the suggestion in the comments I decided to post it here. It is a well-known fact that $$\sum_{0\neq n\...
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Is Brun's constant a period?

Brun proved that the sum of reciprocals of the twin primes is finite. This sum, Brun's constant, is not known to be irrational, (otherwise this would immediately entail the veracity of the twin prime ...
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Comparing an integral to zero, by slicing and stacking

Let $f \colon [0,1] \rightarrow {\mathbb R}$ be a nice function -- real-analytic, or maybe definable in some o-minimal structure, let's say. Let $0 < \alpha_1 < \cdots < \alpha_n < 1$ be ...
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1answer
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Kinds of differentials and algebraic groups

This Wikipedia article mentions that the analogues of differentials of the first/second/third kind for algebraic groups are abelian varieties/algebraic tori/linear algebraic groups. I guess ...
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$l$-adic periods?

For an algebraic variety $X$ over $\mathbb{Q}$ the comparison isomorphism between Betti and de Rham cohomologies provides the theory of periods with a motivic context whose reformulation as motivic ...
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periods of higher weight modular forms

For now let $f\in S_2(\Gamma_0(N))$ and define $I(\alpha,\beta)=\int_\alpha^\beta f(z)dz$. If $M\in\Gamma_0(N)$ it is immediate to prove that $I(\alpha,M(\alpha))$ is independent of $\alpha$: this ...
8
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How to compute cohomology using differentials of the second kind on a Fermat curve?

Differentials of the second kind Gross and Rohrlich in the paper On the periods of abelian integrals and a formula of Chowla and Selberg state the claim below without citation (pg. 198), giving an ...
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1answer
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Higher Genus Surfaces That “Look Like” Genus-1 Surfaces

It is my understanding that a genus-$g$ Riemann surface has $2g$ independent cycles that satisfy the usual intersection rules: $$a_i \cap a_j = 0$$ $$b_i \cap b_j = 0$$ $$a_i \cap b_j = \delta_{ij}$$ ...
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1answer
204 views

Simple integral representation for a beta function with more than two variables

The beta function $B(x,y)=\dfrac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)}$ can be written for $\Re x, \Re y > 0$ symmetrically as $$ B(x,y) = \int_{-\frac12}^{\frac12}(\frac12-t)^{x-1}(\frac12+t)^{y-1}\,...
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Feynman diagrams and periods of motives

A recent article in the online science magazine Quanta, Strange Numbers Found in Particle Collisions, discusses experimental evidence of a connection between Feynman integrals and periods of motives. ...
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1answer
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What is explicitly, $ \mathcal{P} \mathrm{er} ( X ) / \mathbb{Q} $?

In the following link$^{[1]}$, page $2$, we find the following question : Let $X$ be a smooth $ \mathbb{Q} $ - variety and let $\mathcal{P} \mathrm{er} (X)$ be the subfield of $\mathbb{C}$ ...
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Open Period Integrals of Elliptically Fibered K3 surfaces

Let M be the period domain for elliptic K3 surfaces $(X,\Omega)$ with a holomorphic two-form. Denote the fiber class $f$. Then $$M=\{\Omega\in f^\perp\otimes \mathbb{C}\,:\, \Omega\cdot \Omega=0, \,\...
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Can there be a nonzero period integral of this form?

I have been trying to compute the following integral: $$\displaystyle \int_{\gamma} \frac{g \Omega_{\mathbb{P}^3}}{f^2}$$ where: $\gamma$ is the torus cycle $\{ |z_1| =|z_2| =|z_3| =|z_4| =1 \}$, $...
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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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1answer
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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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$\int_0^\infty x \, [J_0(x)]^5 \, dx$: source and context, if any?

QUESTION Numerical calculation with gp (first to the default 38-digit precision, then tripled) supports the conjecture that $$ \int_0^\infty x \, [J_0(x)]^5 \, dx = \frac{\Gamma(1/15) \, \Gamma(2/15) ...
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Computer software for periods

Kontsevich and Zagier define a period as an integral of a rational function (over $\mathbb{Q}$) defined on a $\mathbb{Q}$-semialgebraic set. They conjecture that if two periods are equal, then the ...
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1answer
681 views

Are there integral representations of the Mertens constant?

It is well-known that the Euler constant $$\gamma=\lim\limits_{n\to \infty}\biggl( \sum\limits_{k\le n}\dfrac{1}{k}-\ln{n}\biggr ) $$ has a bunch of integral representations, e.g. $$\gamma=-\int\...
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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 ...
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What is the relationship between these two notions of “period”?

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 ...
33
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1answer
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Special values of L-functions as periods

If $M$ is a pure motive over $\mathbb{Q}$, one cas define its $L$-function $L(M,s)$ which conjecturaly is a meromorphic function over $\mathbb{C}$ with finitely many poles. For example, when $M=\...
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Is it known that the ring of periods is not a field?

I have just learned here that we know numbers that are not periods; is it known meanwhile that the ring of periods is not a field? I know that it is conjectured that $1/\pi$ is not a period, but the ...