Abstract
Several asymptotic expansions and formulas for cubic exponential sums are derived. The expansions are most useful when the cubic coefficient is in a restricted range. This generalizes previous results in the quadratic case and helps to clarify how to numerically approximate cubic exponential sums and how to obtain upper bounds for them in some cases.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
E. Bombieri and H. Iwaniec, “On the order of ζ(\(\frac{1}{2} + it\)),” Ann. Sc. Norm. Super. Pisa, Cl. Sci., Ser. 4, 13 (3), 449–472 (1986).
H. Davenport, Multiplicative Number Theory, 3rd ed., rev. and with a preface by H. L. Montgomery (Springer, New York, 2000), Grad. Texts Math. 74.
H. Fiedler, W. Jurkat, and O. Körner, “Asymptotic expansions of finite theta series,” Acta Arith. 32 (2), 129–146 (1977).
G. H. Hardy and J. E. Littlewood, “Some problems of Diophantine approximation. II: The trigonometrical series associated with the elliptic ϑ-functions,” Acta Math. 37, 193–239 (1914).
G. A. Hiary, “Fast methods to compute the Riemann zeta function,” Ann. Math., Ser. 2, 174 (2), 891–946 (2011).
M. A. Korolev, “On incomplete Gaussian sums,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 290, 61–71 (2015) [Proc. Steklov Inst. Math. 290, 52–62 (2015)].
E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, 2nd ed., rev. by D. R. Heath-Brown (Clarendon Press, Oxford, 1986).
Author information
Authors and Affiliations
Corresponding author
Additional information
Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2017, Vol. 299, pp. 86–104.
Rights and permissions
About this article
Cite this article
Hiary, G.A. Asymptotics and Formulas for Cubic Exponential Sums. Proc. Steklov Inst. Math. 299, 78–95 (2017). https://doi.org/10.1134/S0081543817080053
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0081543817080053