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The Riemann zeta function at integer arguments can be written as an infinitesum of certain hypergeometric functions and more generally the same can be donewith polylogarithms, for which several zeta functions are a special case. Ananalytic continuation formula for these hypergeometric functions exists and isused to derive some infinite sums which allow the zeta function at integerarguments n to be written as a weighted infinite sum of hypergeometricfunctions at n - 1. The form might be considered to be a shift operator for theRiemann 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 thefirst kind and an exponential integral respectively and differ from the values{\zeta}(0) = -1/2 and {\zeta}(1) = \infty given by the usual method ofcontinuation. Interpreting these "hypergeometrically continued" values of thezeta 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 withthe 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 Stirlingnumbers of the second kind, and the 2-restricted Stirling numbers of the secondkind are recalled because they appear in the differential equatlon satisfied bythe hypergeometric representation of the polylogarithm. The notion of fractalstrings is related to the (chaotic) Gauss map of the unit interval which arisesin the study of continued fractions, and another chaotic map is also introducedcalled the "Harmonic sawtooth" whose Mellin transform is the (appropritatelyscaled) Riemann zeta function. These maps are within the family of what mightbe called "deterministic chaos". Some number theoretic definitions are alsorecalled.

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