## The Approximate Functional Formula for the Theta Function and Diophantine Gauss Sums

Please use this identifier to cite or link to this item: http://hdl.handle.net/1928/20185

Title

The Approximate Functional Formula for the Theta Function and Diophantine Gauss Sums

Author(s)

Coutsias, Evangelos A.

Kazarinoff, N.D.

Abstract

By introducing the discrete curvature of the polygonal line, and by exploiting the similarity of segments of the line, for small w, to Cornu spirals (C-spirals), we prove the precise renormalization formula. This formula, which sharpens Hardy and Littlewood's approximate functional formula for the theta function, generalizes to irrationals, as a Diophantine inequality, the well-known sum formula of Gauss. The geometrical meaning of the relation between the two limits is that the first sum is taken to a point of inflection of the corresponding C-spirals. The second sum replaces whole C-spirals of the first by unit vectors times scale and phase factors. The block renormalization procedure implied by this replacement is governed by the circle map whose orbits are analyzed by expressing w as an even continued fraction.

Publisher

American Mathematical Society

Citation

Transactions of the American Mathematical Society, 350(2): 615-641

ISSN

0002-9947