Central Discovery

One of the most interesting discoveries in the research that spins around the function S(n) that I discovered in 1964 is its ability to take the square root of the Zeta.  For example the value of the Zeta Function for an exponent of 2 is π2/6.  as proven by Leonard Euler in 1735.  My function S(n) when used in conjunction with the Zeta takes the square root of the zeta which is for exponet 2 = π/√6.   This theorem applies to all real and complex exponents.  I have a beautiful proof for this fact.

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Here is the Mathematica program that illustrates the theorem, but of ourse does not prove it.  That is done formally.

(* 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

(*** Begin your work here ***)

t1 = Table[SSN[i]/i^2,{i,1,10000}];

mytotal =Total[t1];

Print["S(n) takes the square root ",N[mytotal,5]];

root = Sqrt[Zeta[2]];

Print["Mathematica's Square Root = ",N[root,5]]