Two New Zeta Constants: Fractal String, Continued Fraction, and Hypergeometric Aspects of the Riemann Zeta Function

Abstract

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.