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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The approximate functional formula for the theta function and Diophantine Gauss sums

Authors: E. A. Coutsias and N. D. Kazarinoff
Journal: Trans. Amer. Math. Soc. 350 (1998), 615-641
MSC (1991): Primary 11G10; Secondary 11L05, 11L07, 11J25, 11J70
MathSciNet review: 1443869
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Abstract: We consider the polygonal lines in the complex plane $\Bbb {C}$ whose $N$-th vertex is defined by $S_N = \sum _{n=0}^{N ’} \exp (i\omega \pi n^2)$ (with $\omega \in \Bbb {R}$), where the prime means that the first and last terms in the sum are halved. By introducing the discrete curvature of the polygonal line, and by exploiting the similarity of segments of the line, for small $\omega$, to Cornu spirals (C-spirals), we prove the precise renormalization formula \begin{equation} \begin {split} &\left | \sum _{k=0}^{N} ’ \exp (i\omega \pi k^2) -\frac {\exp (sgn(\omega )i\pi /4)}{\sqrt {|\omega |}} \sum _{k=0}^n ’ \exp (-i\frac {\pi }{\omega } k^2)\right |\\ &\qquad \leq C \left |\frac {\omega N - n}{\omega }\right |, 0<|\omega | <1, \end{split} \end{equation} where $N=[[n/\omega ]]$, the nearest integer to $n/\omega$ and $1<C<3.14$ . 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 \begin{equation} \omega \rightarrow -\frac {1}{\omega } \pmod 2 , \omega \in ]-1,+1[ \setminus \{0\}, \end{equation} whose orbits are analyzed by expressing $\omega$ as an even continued fraction.

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  • Tom M. Apostol, Introduction to analytic number theory, Springer-Verlag, New York-Heidelberg, 1976. Undergraduate Texts in Mathematics. MR 0434929
  • M. V. Berry and J. Goldberg, Renormalisation of curlicues, Nonlinearity 1 (1988), no. 1, 1–26. MR 928946
  • M. V. Berry, Random renormalization in the semiclassical long-time limit of a precessing spin, Phys. D 33 (1988), no. 1-3, 26–33. Progress in chaotic dynamics. MR 984607, DOI
  • J.L. Callot and M. Diener, “Variations en spirale”, Document du travail 6, p. 16-50, Oran, 1984.
  • K. Chandrasekharan, Elliptic functions, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 281, Springer-Verlag, Berlin, 1985. MR 808396
  • J.G. van der Corput, Über Summen die mit den Elliptischen $\theta$-Funktionen zusammenhägen, I, II, Math. Ann. 87, (1922), 66-77; 90, (1923), 1-18.
  • J.G. van der Corput, Beweis einer approximativen Funktionalgleichung, Mathematische Zeitschrift, 28 (1928) 238-300.
  • Evangelos A. Coutsias and Nicholas D. Kazarinoff, Disorder, renormalizability, theta functions and Cornu spirals, Phys. D 26 (1987), no. 1-3, 295–310. MR 892449, DOI
  • F. M. Dekking and M. Mendès France, Uniform distribution modulo one: a geometrical viewpoint, J. Reine Angew. Math. 329 (1981), 143–153. MR 636449, DOI
  • Jean-Marc Deshouillers, Geometric aspect of Weyl sums, Elementary and analytic theory of numbers (Warsaw, 1982) Banach Center Publ., vol. 17, PWN, Warsaw, 1985, pp. 75–82. MR 840473
  • G. H. Hardy and J. E. Littlewood, Some problems of Diophantine approximation, Acta Math. 37, (1914), 193-238.
  • Peter Henrici, Applied and computational complex analysis. Vol. 2, Wiley Interscience [John Wiley & Sons], New York-London-Sydney, 1977. Special functions—integral transforms—asymptotics—continued fractions. MR 0453984
  • Joseph B. Keller and Charles Knessl, Asymptotic evaluation of oscillatory sums, European J. Appl. Math. 4 (1993), no. 4, 361–379. MR 1251820, DOI
  • N. M. Korobov, Exponential sums and their applications, Mathematics and its Applications (Soviet Series), vol. 80, Kluwer Academic Publishers Group, Dordrecht, 1992. Translated from the 1989 Russian original by Yu. N. Shakhov. MR 1162539
  • L. Kronecker, Summierung der Gaussschen Reihen $\sum _{h=0}^{h=n-1}e^{2h^2\pi i/n}$, Journal für die reine und angewandte Mathematik, 105(1889), 267-268.
  • D. H. Lehmer, Incomplete Gauss sums, Mathematika 23 (1976), no. 2, 125–135. MR 429787, DOI
  • J. H. Loxton, The graphs of exponential sums, Mathematika 30 (1983), no. 2, 153–163 (1984). MR 737174, DOI
  • Jacques Bélair and Serge Dubuc (eds.), Fractal geometry and analysis, NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, vol. 346, Kluwer Academic Publishers Group, Dordrecht, 1991. MR 1140718
  • Hugh L. Montgomery, Ten lectures on the interface between analytic number theory and harmonic analysis, CBMS Regional Conference Series in Mathematics, vol. 84, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1994. MR 1297543
  • R. R. Moore and A. J. van der Poorten, On the thermodynamics of curves and other curlicues, Miniconference on Geometry and Physics (Canberra, 1989) Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 22, Austral. Nat. Univ., Canberra, 1989, pp. 82–109. MR 1027862
  • L.J. Mordell, The approximate functional formula for the theta function, J. London Mat. Soc. 1(1926) 68-72.
  • I.M. Vinogradov, The Method of Trigonometric Sums in the Theory of Numbers, translated from the Russian, revised and annotated by K.F. Roth and A. Davenport (Interscience, London, 1954).
  • J. R. Wilton, The approximate functional formula for the theta function, J. London Math. Soc. 2 (1926), 177-180.

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Additional Information

E. A. Coutsias
Affiliation: Department of Mathematics and Statistics, University of New Mexico, Albuquerque, New Mexico 87131

N. D. Kazarinoff
Affiliation: Department of Mathematics and Statistics, University of New Mexico, Albuquerque, New Mexico 87131

Received by editor(s): January 25, 1995
Received by editor(s) in revised form: December 4, 1995
Article copyright: © Copyright 1998 American Mathematical Society