Riemann Zeta Function. An Attempt to fathom Zeta(3)

Abstract

Already in 1734 Euler found a short explicit formula for the value of Riemann zeta function Zeta(s) when the argument s equals a positive integer 2n where n=1,2,3,. No such formula exists for odd positive integer arguments of Zeta. The present paper discusses in particular the case of Zeta(3). A formula for Zeta(3) is obtained which in addition to a number of well known constants includes a rapidly converging infinite series, of which each term contains rational numbers and an even power of Pi. An attempt to convert this series into a finite number of terms containing commonly known constants is met with only partial success. The general case for zeta(2n+1) is also worked out.