Zeta2(s1,s2),Zeta3(s1,s2,s3):Integral Expressions and Approximates

TL;DR

This paper derives integral representations and explicit approximations for the multiple zeta functions zeta2 and zeta3, facilitating their computation especially when variables differ by even integers or when s2 is large.

Contribution

It provides new integral formulas and explicit approximations for zeta2 and zeta3, enhancing their computational accessibility.

Findings

Integral representation involving Hurwitz zeta functions

Explicit approximation when s1,s2 differ by an even integer

Good approximation for large s2

Abstract

For the multiple zeta function zeta2(s1,s2) of two variables,we obtain its integral representation(involving product of Hurwitz zeta functions) over the interval [1,infinity),with respect to second variable of Hurwitz zeta function and also obtain a good approximate to it as a function of s1 and s2,for s1>=1 and s2>1.In particular this approximate is explicitly computable,when s1,s2 differ by an even integer and is good,when s2 is large.We treat zeta3(s1,s2,s3) likewise.