Hostname: page-component-76fb5796d-22dnz Total loading time: 0 Render date: 2024-04-27T01:21:38.712Z Has data issue: false hasContentIssue false
  • English
  • Français

THE VALUE DISTRIBUTION OF INCOMPLETE GAUSS SUMS

Part of: Exponential sums and character sums

Published online by Cambridge University Press:  19 February 2013

Emek Demirci Akarsu  and
Jens Marklof
Emek Demirci Akarsu*
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW, U.K.
Jens Marklof*
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW, U.K.
*
E.DemirciAkarsu@bristol.ac.uk
j.marklof@bristol.ac.uk
Get access

Abstract

It is well known that the classical Gauss sum, normalized by the square-root number of terms, takes only finitely many values. If one restricts the range of summation to a subinterval, a much richer structure emerges. We prove a limit law for the value distribution of such incomplete Gauss sums. The limit distribution is given by the distribution of a certain family of periodic functions. Our results complement Oskolkov’s pointwise bounds for incomplete Gauss sums as well as the limit theorems for quadratic Weyl sums (theta sums) due to Jurkat and van Horne and the second author.

MSC classification

Secondary: 11L05: Gauss and Kloosterman sums; generalizations
Type
Research Article
Information
Mathematika , Volume 59 , Issue 2 , July 2013 , pp. 381 - 398
Copyright
Copyright © University College London 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arkhipov, G. I. and Oskolkov, K. I., A special trigonometric series and its applications. Sb. Math. 62 (1989), 145–155.CrossRefGoogle Scholar
Cellarosi, F., Limiting curlicue measures for theta sums. Ann. Inst. Henri Poincaré Probab. Stat. 47 (2011), 466–497.CrossRefGoogle Scholar
Chinen, K., On estimation of Kloosterman sums with the theta multiplier. Math. Jpn 48 (1998), 223–232.Google Scholar
Duke, W., Friedlander, J. B. and Iwaniec, H., Weyl sums for quadratic roots. Int. Math. Res. Not. IMRN 2012 (2012), 2493–2549; doi:10.1093/imrn/rnr112.Google Scholar
Estermann, T., On Kloosterman’s sum. Mathematika 8 (1961), 83–86.CrossRefGoogle Scholar
Evans, R., Minei, M. and Yee, B., Incomplete higher order Gauss sums. J. Math. Anal. Appl. 281 (2003), 454–476.CrossRefGoogle Scholar
Fiedler, H., Jurkat, W. B. and Körner, O., Asymptotic expansions of finite theta series. Acta Arith. 32 (1977), 129–146.CrossRefGoogle Scholar
Jurkat, W. B. and van Horne, J. W., The proof of the central limit theorem for theta sums. Duke Math. J. 48 (1981), 873–885.CrossRefGoogle Scholar
Jurkat, W. B. and van Horne, J. W., On the central limit theorem for theta series. Michigan Math. J. 29 (1982), 65–67.CrossRefGoogle Scholar
Jurkat, W. B. and van Horne, J. W., The uniform central limit theorem for theta sums. Duke Math. J. 50 (1983), 649–666.CrossRefGoogle Scholar
Lehmer, D. H., Incomplete Gauss sums. Mathematika 23 (1976), 125–135.CrossRefGoogle Scholar
Marklof, J., Limit theorems for theta sums. Duke Math. J. 97 (1999), 127–153.CrossRefGoogle Scholar
Montgomery, H. L., Vaughan, R. C. and Wooley, T. D., Some remarks on Gauss sums associated with kth powers. Math. Proc. Cambridge Philos. Soc. 118 (1995), 21–33.CrossRefGoogle Scholar
Oskolkov, K. I., On functional properties of incomplete Gaussian sums. Can. J. Math. 43 (1991), 182–212.CrossRefGoogle Scholar
Paris, R. B., An asymptotic approximation for incomplete Gauss sums. J. Comput. Appl. Math. 180 (2005), 461–477.CrossRefGoogle Scholar
Paris, R. B., An asymptotic approximation for incomplete Gauss sums, II. J. Comput. Appl. Math. 212 (2008), 16–30.CrossRefGoogle Scholar
Shimura, G., On modular forms of half integral weight. Ann. of Math. (2) 97 (1973), 440–481.CrossRefGoogle Scholar