  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.
