Euler Sums Revisited

A large literature exists relating Riemann’s zeta function ζ(s)=∑k=1 ∞ 1/ks,R(s)>1, and partial sums of the harmonic series, h(n)=∑k=1 n 1/k. Much of the research originated from two striking formulas discovered by Euler, ∑n=1 ∞ h(n)/n^2=2ζ(3) (1) , ∑n=1 ∞ h(n)/n^3=54ζ(4), (2) and a recursion formula, also due to Euler, which states that for integer a ≥2 we have (a+1/2)ζ(2a)=∑k=1 a−1 ζ(2k)ζ(2a−2k). (3) For a = 2 and a =3 this gives ζ(4) =2/5ζ(2)^2 and ζ(6) =8/35ζ(2)^3. More generally, it shows that ζ(2) is a rational multiple of ζ(2) n. These results were rediscovered and extended by Ramanujan [11] and many others [1][5][6][8]