  The Riemann zeta function at integer arguments can be written as an infinite
sum of certain hypergeometric functions and more generally the same can be done
with polylogarithms, for which several zeta functions are a special case. An
analytic continuation formula for these hypergeometric functions exists and is
used to derive some infinite sums which allow the zeta function at integer
arguments n to be written as a weighted infinite sum of hypergeometric
functions at n - 1. The form might be considered to be a shift operator for the
Riemann zeta function which leads to the curious values {\zeta}F(0) = I_0(2) -
1 and {\zeta}F(1) = Ei(1) - {\gamma} which involve a Bessel function of the
first kind and an exponential integral respectively and differ from the values
{\zeta}(0) = -1/2 and {\zeta}(1) = \infty given by the usual method of
continuation. Interpreting these "hypergeometrically continued" values of the
zeta constants in terms of reciprocal common factor probability we have
{\zeta}F(0)^-1 \sim 78.15% and {\zeta}F(1)^-1 \sim 75.88% which contrasts with
the standard known values for sensible cases like {\zeta}(2)^-1 \sim 60.79% and
{\zeta}(3)^-1 \sim 83.19%. The combinatorial definitions of the Stirling
numbers of the second kind, and the 2-restricted Stirling numbers of the second
kind are recalled because they appear in the differential equatlon satisfied by
the hypergeometric representation of the polylogarithm. The notion of fractal
strings is related to the (chaotic) Gauss map of the unit interval which arises
in the study of continued fractions, and another chaotic map is also introduced
called the "Harmonic sawtooth" whose Mellin transform is the (appropritately
scaled) Riemann zeta function. These maps are within the family of what might
be called "deterministic chaos". Some number theoretic definitions are also
recalled.
