Functional Equations of Tornheim Zeta Function and the alpha-calculus of r-th order derivative of Zeta(s,alpha)

TL;DR

This paper derives new functional equations for the Tornheim double zeta function using the alpha-calculus of the r-th order derivative of the zeta function, enabling explicit evaluations at specific points and analogues of classical formulas.

Contribution

It introduces novel functional equations for Tornheim zeta functions and applies the alpha-calculus to evaluate these functions at various complex and integer points.

Findings

Derived new functional equations for Tornheim zeta function.

Evaluated T(n,n,n) for any complex s and integers n>=1.

Established analogues of Euler's formulas for Tornheim zeta functions.

Abstract

For the Tornheim double zeta function T(s1,s2,s3) of complex variables,we obtain its functional equations,which are new.Using the calculus of r-th order derivative of zeta(s,alpha) as a function of alpha(developed in author[7])as the tool,we obtain with ease an expression for T(s,s,s) for any complex s;and evaluate T(n,n,n) for any integer n>=1;and also evaluate T(n1,0,n2)for integers n1,n2>1 with n1+n2 odd.We obtain for Tornheim zeta function,the counterparts of Euler's formula for zeta(2n)and its analogue for zeta(2n+1),where Zeta(s) is Riemann zeta function.