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==
1SSN[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]]