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

The Riemann zeta function at integer arguments can be written as an infinite\nsum of certain hypergeometric functions and more generally the same can be done\nwith polylogarithms, for which several zeta functions are a special case. An\nanalytic continuation formula for these hypergeometric functions exists and is\nused to derive some infinite sums which allow the zeta function at integer\narguments n to be written as a weighted infinite sum of hypergeometric\nfunctions at n - 1. The form might be considered to be a shift operator for the\nRiemann zeta function which leads to the curious values {\\zeta}F(0) = I_0(2) -\n1 and {\\zeta}F(1) = Ei(1) - {\\gamma} which involve a Bessel function of the\nfirst kind and an exponential integral respectively and differ from the values\n{\\zeta}(0) = -1/2 and {\\zeta}(1) = \\infty given by the usual method of\ncontinuation. Interpreting these "hypergeometrically continued" values of the\nzeta constants in terms of reciprocal common factor probability we have\n{\\zeta}F(0)^-1 \\sim 78.15% and {\\zeta}F(1)^-1 \\sim 75.88% which contrasts with\nthe standard known values for sensible cases like {\\zeta}(2)^-1 \\sim 60.79% and\n{\\zeta}(3)^-1 \\sim 83.19%. The combinatorial definitions of the Stirling\nnumbers of the second kind, and the 2-restricted Stirling numbers of the second\nkind are recalled because they appear in the differential equatlon satisfied by\nthe hypergeometric representation of the polylogarithm. The notion of fractal\nstrings is related to the (chaotic) Gauss map of the unit interval which arises\nin the study of continued fractions, and another chaotic map is also introduced\ncalled the "Harmonic sawtooth" whose Mellin transform is the (appropritately\nscaled) Riemann zeta function. These maps are within the family of what might\nbe called "deterministic chaos". Some number theoretic definitions are also\nrecalled.\n