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25 Zeta and Related FunctionsRelated Functions

§25.13 Periodic Zeta Function

The notation F(x,s) is used for the polylogarithm Lis(e2πix) with x real:

25.13.1 F(x,s)n=1e2πinxns,

where s>1 if x is an integer, s>0 otherwise.

F(x,s) is periodic in x with period 1, and equals ζ(s) when x is an integer. Also,

25.13.2 F(x,s)=Γ(1-s)(2π)1-s(eπi(1-s)/2ζ(1-s,x)+eπi(s-1)/2ζ(1-s,1-x)),
0<x<1, s>1,
25.13.3 ζ(1-s,x)=Γ(s)(2π)s(e-πis/2F(x,s)+eπis/2F(-x,s)),
0<x<1, s>0.