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25 Zeta and Related FunctionsRiemann Zeta Function

§25.9 Asymptotic Approximations

If x1, y1, 2πxy=t, and 0σ1, then as t with σ fixed,

25.9.1 ζ(σ+it)=1nx1ns+χ(s)1ny1n1-s+O(x-σ)+O(yσ-1t12-σ),

where s=σ+it and

25.9.2 χ(s)=πs-12Γ(12-12s)/Γ(12s).

If σ=12, x=y=t/(2π), and m=x, then (25.9.1) becomes

25.9.3 ζ(12+it)=n=1m1n12+it+χ(12+it)n=1m1n12-it+O(t-1/4).

For other asymptotic approximations see Berry and Keating (1992), Paris and Cang (1997); see also Paris and Kaminski (2001, pp. 380–389).