arXiv 1401.78432014-01-30math-phmath.MP

随机Aharonov-Bohm涡与某些精确积分族：III部分

Random Aharonov-Bohm vortices and some exact families of integrals: Part III

Stephane Ouvry

备注：: 13 pages. arXiv admin note: substantial text overlap with arXiv:1209.1036
发表信息：: 2013 J. Stat. Mech. 2005 P09004

摘要：

作为[1]和[2]的续篇，本文介绍了Bessel积分$\int_0^{\infty}{\rmd u}\; uK_0(u)^{n}$，$\int_0^{\infty}{\rmd u}\; u^{3}K_0(u)^{n}$，.其中积分变量的幂是奇数，并且其中贝塞尔权重$n$是正整数。已知权重n=3和n=4的其中一些积分与zeta数zeta（2）和zeta（3）密切相关。从费曼图的启发表示的n维多重积分在一个无限的区域，一个显示如何部分集成到n-2维多重积分在有限的区域。在这个过程中，贝塞尔积分被证明是周期。有趣的是，这些“简化”的多重积分可以与zeta数的一些简单积分表示并行考虑。本文还将文[2]关于二重套Bessel积分的特殊和的构造推广到整个二重套积分族。最后，一个强有力的PSLQ数值证据表明，支持一个令人惊讶的简单表达式zeta（5）作为一个线性组合的合理系数的贝塞尔积分的重量n= 8。

英文摘要：

As a sequel to [1] and [2], I present some recent progress on Bessel integrals $\int_0^{\infty}{\rmd u}\; uK_0(u)^{n}$, $\int_0^{\infty}{\rmd u}\; u^{3}K_0(u)^{n}$, ... where the power of the integration variable is odd and where $n$, the Bessel weight, is a positive integer. Some of these integrals for weights n=3 and n=4 are known to be intimately related to the zeta numbers zeta(2) and zeta(3). Starting from a Feynman diagram inspired representation in terms of n dimensional multiple integrals on an infinite domain, one shows how to partially integrate to n-2 dimensional multiple integrals on a finite domain. In this process the Bessel integrals are shown to be periods. Interestingly enough, these "reduced" multiple integrals can be considered in parallel with some simple integral representations of zeta numbers. One also generalizes the construction of [2] on a particular sum of double nested Bessel integrals to a whole family of double nested integrals. Finally a strong PSLQ numerical evidence is shown to support a surprisingly simple expression of zeta(5) as a linear combination with rational coefficients of Bessel integrals of weight n= 8.

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