Random Aharonov–Bohm vortices and some exact families of integrals: part III

As a sequel to Ouvry (2005 J. Stat. Mech. P09004) and Mashkevich and Ouvry (2008 J. Stat. Mech. P03018), I present some recent progress on Bessel integrals 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 4 are known to be intimately related to the zeta numbers ζ(2) and ζ(3). Starting from a Feynman diagram inspired representation in terms of an n-dimensional multiple integral on an infinite domain, I show how to partially integrate to an n − 2 multiple integral on a finite domain. In this process the Bessel integrals are shown to be periods. Interestingly, these 'reduced' multiple integrals can be considered in parallel with some simple integral representations of ζ numbers. I also generalize the construction made for a particular sum of double-nested Bessel integrals to a whole family of double-nested integrals. Finally strong PSLQ numerical evidence is shown to support a surprisingly simple expression for ζ(5) as a linear combination with rational coefficients of Bessel integrals of weight n = 8. (paper)