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The Riemann-Siegel formula and large scale computations of the Riemann zeta function

University of British Columbia

This thesis is a survey of the derivation and implementation of the Riemann-Siegel formula for computing values of Riemann's zeta function on the line s = 1/2 + it. The formula, devised by Riemann and later published by Siegel following study of Riemann's unpublished work, is the method of choice for both numerically verifying the Riemann Hypothesis and locating zeros on the critical line at large values of t. Simply stated, the Riemann Hypothesis is that all of the zeros of ζ(s) in the strip 0 < R(s) < 1 lie on the line R(s) = 1/2. Since Riemann made his conjecture in 1859, much work has been done towards developing efficient numerical techniques for verifying the hypothesis and possibly finding counter-examples. This thesis is meant to serve as a guide book for using Riemann-Siegel. It is mostly a distillation of work done in the field since Siegel's results published in the early 1930's. Computer programs and examples are included, and error bounds are discussed. The question of how and why Riemann-Siegel is used to verify the Riemann Hypothesis is examined, and a detailed Riemann Hypothesis verification example is illustrated. Finally, recent work in the field is noted. The derivation of the Riemann-Siegel formula for computing ζ(l/2 + it) is based on the saddle point method of evaluating integrals, and yields results of considerable accuracy in time t1/2. The saddle point method is an approximation technique which concentrates the "bulk" of an integral on a path through a point at which the modulus of the integrand is a maximum. [Mathematical formulae used in this abstract could not be reproduced.]

Thesis/Dissertation

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2009-06-12

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http://hdl.handle.net/2429/9023

Mathematics

University of British Columbia

UBCV

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