International Journal of Mathematics and M Computer Science, 7(2012), no. 1, 11–83 CS Some infinite series involving the Riemann zeta function Donal F. Connon Elmhurst Dundle Road Matfield, Kent TN12 7HD, United Kingdom email:
[email protected] (Received May 16, 2010, Accepted July 12, 2012) Abstract This paper considers some infinite series involving the Riemann zeta function. Some examples are set out below ∞ n n − kxk 1 ( 1) s x, s ,y − x x, s, y n k k y s = Φ( +1 ) log Φ( ) n=0 +1k=0 ( + ) ∞ n n k − n 2 1 − 1−s ζ s ( 1) k k s = 1 2 ( ) n=0 k=0 ( +1) 2 ∞ n n − k − k 1 ( 1) ( 1) π πy n+1 k k y + k − y = cosec( ) n=0 2 k=0 + +1 ∞ 2n 2n u n −1 2 sin 2 u2 u u u u − ζ n3 n =4 log(2 sin )+2Cl3(2 )+4 Cl2(2 ) 2 (3) n=1 ∞ 4 7 1 24n [n!] ζ(3) − πG = n 3 n 2 4 2 n=0 (2 +1) [(2 )!] where Φ(x, s, y) is the Hurwitz-Lerch zeta function and Cln(t) are the Clausen functions. Key words and phrases: Infinite series, Riemann zeta Function. AMS (MOS) Subject Classifications: 40A05, 11M32. 12 Donal F. Connon 1 Some Hasse-type series It was shown in [21] that ∞ n k k y 1 n (−1) y log y = Lis(y) − Lis−1(y)(1) − k s−1 − s 1 n=0 n +1 k=0 (k +1) s 1 where Lis(y) is the polylogarithm function [39] ∞ yn Lis(y)= s n=1 n and, with y = 1, this becomes the formula originally discovered by Hasse [32] in 1930 ∞ n k 1 1 n (−1) = Lis(1) = ζ(s)(2) − k s−1 s 1 n=0 s +1 k=0 (k +1) ∞ s−1 −t The gamma function is defined as Γ(s)= 0 t e dt, s>0.