Riemann-Siegel function theta: Series representations (formula 10.03.06.0025)

RiemannSiegelTheta

$Mathematica Notation$

Zeta Functions and Polylogarithms

RiemannSiegelTheta[z]

Series representations

Generalized power series

Expansions on branch cuts

In the upper half-plane

http://functions.wolfram.com/10.03.06.0025.01

Input Form

RiemannSiegelTheta[z] \[Proportional] RiemannSiegelTheta[Subscript[z, 0]] + Pi Floor[Arg[I (z - Subscript[z, 0])]/(2 Pi)] Floor[(1 + 2 I Subscript[z, 0])/4] - (1/4) (2 Log[Pi] - PolyGamma[1/4 - (I Subscript[z, 0])/2] - PolyGamma[1/4 + (I Subscript[z, 0])/2]) (z - Subscript[z, 0]) - (I/16) (PolyGamma[1, 1/4 - (I Subscript[z, 0])/2] - PolyGamma[1, 1/4 + (I Subscript[z, 0])/2]) (z - Subscript[z, 0])^2 + \[Ellipsis] /; (z -> Subscript[z, 0]) && Element[I Subscript[z, 0], Reals] && I Subscript[z, 0] < -(1/2) && !Element[(I Subscript[z, 0])/2, Integers]

Standard Form

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MathML Form

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Rule Form

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Date Added to functions.wolfram.com (modification date)

2007-05-02