Algebra & Number Theory

Multiplicative mimicry and improvements to the Pólya–Vinogradov inequality

Leo Goldmakher

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We study exponential sums whose coefficients are completely multiplicative and belong to the complex unit disc. Our main result shows that such a sum has substantial cancellation unless the coefficient function is essentially a Dirichlet character. As an application we improve current bounds on odd-order character sums. Furthermore, conditionally on the generalized Riemann hypothesis we obtain a bound for odd-order character sums which is best possible.

Article information

Algebra Number Theory, Volume 6, Number 1 (2012), 123-163.

Received: 21 October 2010
Accepted: 29 December 2010
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11L40: Estimates on character sums
Secondary: 11L03: Trigonometric and exponential sums, general 11L07: Estimates on exponential sums

Dirichlet characters character sums exponential sums multiplicative functions


Goldmakher, Leo. Multiplicative mimicry and improvements to the Pólya–Vinogradov inequality. Algebra Number Theory 6 (2012), no. 1, 123--163. doi:10.2140/ant.2012.6.123.

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  • A. Balog, A. Granville, and Soundararajan, “Multiplicative functions in arithmetic progressions”, preprint, 2007. To appear in Ann. Sci. Math. Québec.
  • A. Granville and K. Soundararajan, “The distribution of values of $L(1,\chi\sb d)$”, Geom. Funct. Anal. 13:5 (2003), 992–1028.
  • A. Granville and K. Soundararajan, “Large character sums: pretentious characters and the Pólya-Vinogradov theorem”, J. Amer. Math. Soc. 20:2 (2007), 357–384.
  • A. Granville and K. Soundararajan, “Pretentious multiplicative functions and an inequality for the zeta-function”, pp. 191–197 in Anatomy of integers, edited by J.-M. De Koninck et al., CRM Proc. Lecture Notes 46, Amer. Math. Soc., Providence, RI, 2008.
  • G. Halász, “On the distribution of additive and the mean values of multiplicative arithmetic functions”, Studia Sci. Math. Hungar. 6 (1971), 211–233.
  • A. Hildebrand, “Large values of character sums”, J. Number Theory 29:3 (1988), 271–296.
  • J. E. Littlewood, “On the class number of the corpus $P(\sqrt{-k})$”, Proc. London Math. Soc. 27 (1928), 358–372.
  • H. L. Montgomery, “A note on mean values of multiplicative functions”, report 17, Institute Mittag-Leffler, Djursholm, 1978.
  • H. L. Montgomery and R. C. Vaughan, “Exponential sums with multiplicative coefficients”, Invent. Math. 43:1 (1977), 69–82.
  • H. L. Montgomery and R. C. Vaughan, “Mean values of multiplicative functions”, Period. Math. Hungar. 43:1-2 (2001), 199–214. 0980.11043
  • R. Paley, “A theorem on characters”, J. London Math. Soc. 7:1 (1932), 28–32. 58.0192.01
  • G. Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge Studies in Advanced Mathematics 46, Cambridge University Press, 1995.
  • S. V. Vostokov and K. Y. Orlova, “Generalization and application of the Eisenstein reciprocity law”, Vestnik St. Petersburg Univ. Math. 41:1 (2008), 15–20. 2009c:11007