Given integers 0 =< r < k, is there a standard way to relate the series
Z(s; k,r) := Sum{n=1..oo} 1/(kn+r)^s
to
Zeta(s) := Sum{n=1..oo} 1/n^s
?
(Of course for r = 0 this is trivial.)
--Dan
No, you need Hurwitz zeta or polygammas: (c9) (sum((k+1/3)^-2,k,0,inf),%% = closedform(%%)) inf ==== \ 1 1 (d9) > -------- = Psi (-) / 1 2 1 3 ==== (k + -) k = 0 3 (c10) (sum((k+1/4)^-2,k,0,inf),%% = closedform(%%)) inf ==== \ 1 2 (d10) > -------- = %pi + 8 %catalan / 1 2 ==== (k + -) k = 0 4 (c11) (sum((k+1/3)^-3,k,0,inf),%% = closedform(%%)) inf ==== 3 \ 1 2 %pi (d11) > -------- = 13 zeta(3) + --------- / 1 3 3 sqrt(3) ==== (k + -) k = 0 3 (c12) (sum((k+1/4)^-3,k,0,inf),%% = closedform(%%)) inf ==== \ 1 3 (d12) > -------- = 28 zeta(3) + %pi / 1 3 ==== (k + -) k = 0 4 Look, Ma, no Yahoo spamdangles.