Riemann’s Saddle-point Method and the Riemann-Siegel Formula
نویسنده
چکیده
Riemann’s way to calculate his zeta function on the critical line was based on an application of his saddle-point technique for approximating integrals that seems astonishing even today. His contour integral for the remainder in the Dirichlet series for the zeta function involved not an isolated saddle, nor a saddle near a pole or an end-point or several coalescing saddles, but the configuration, unfamiliar even now, of a saddle close to an infinite string of poles. Riemann evaluated the associated integral exactly, and the resulting Riemann-Siegel formula underlies ways of computing the Riemann zeros and one of the physical approaches to the Riemann hypothesis. 2000 Mathematics Subject Classification: 01A55, 11M26, 11Y35, 30B50, 30E15, 34E05, 41A60.
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