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25 Zeta and Related FunctionsRiemann Zeta Function

§25.8 Sums

25.8.1 k=2(ζ(k)-1)=1.
25.8.2 k=0Γ(s+k)(k+1)!(ζ(s+k)-1)=Γ(s-1),
s1,0,-1,-2,.
25.8.3 k=0Γ(s+k)ζ(s+k)k!Γ(s)2s+k=(1-2-s)ζ(s),
s1.
25.8.4 k=1(-1)kk(ζ(nk)-1)=ln(j=0n-1Γ(2-e(2j+1)πi/n)),
n=2,3,4,.
25.8.5 k=2ζ(k)zk =-γz-zψ(1-z),
|z|<1.
25.8.6 k=0ζ(2k)z2k =-12πzcot(πz),
|z|<1.
25.8.7 k=2ζ(k)kzk =-γz+lnΓ(1-z),
|z|<1.
25.8.8 k=1ζ(2k)kz2k =ln(πzsin(πz)),
|z|<1.
25.8.9 k=1ζ(2k)(2k+1)22k=12-12ln2.
25.8.10 k=1ζ(2k)(2k+1)(2k+2)22k=14-74π2ζ(3).

For other sums see Prudnikov et al. (1986b, pp. 648–649), Hansen (1975, pp. 355–357), Ogreid and Osland (1998), and Srivastava and Choi (2001, Chapter 3).