Definition of S(n) and Table of Values from 1 to 100

Table of Values of S(n) from 1 to 100

S(n) = (1 - ∑S(d)S(n/d))/R(n)

d|n

 (d,n/d)≠1

The vlaues in the table below come from a multiplicative function that I call S(n).  It has amazing properties.  As you can see from the table, S(n) is multiplicative.  This function can be used in the numerator of the Riemann Zeta Function.  When so used, S(n) takes the square root of the Zeta in both the real and complex number system.

The Mathematica Program to Produce any value of S(n) Follows:

(* This Function computes the value of S(p^n) *)

Clear[SofP]

(* SofP[n_] :=((2n)!/(n!*n!))(1/2)^(2*n) is the factorial definition *)

(* What follows is the Gamma Definition -- any n works here *)

SofP[n_]:=(Gamma[2*n+1]/Gamma[n+1]^2)(1/2)^(2*n)

(* The Function SSN computes S(n) for any positive integer n *)

Clear[SSN]

SSN[n_]:= 1 /; n==1

SSN[n_]:=Product[SofP[Part[FactorInteger[Floor[n]],j,2]],{j,1, Length[FactorInteger[Floor[n]]]}] /;Abs[n-Floor[n]]<= .00001

SSN[n_]:=SofP[n]/;Abs[n-Floor[n]]> .00001

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