Multiplicative mimicry and improvements of the Polya-Vinogradov inequality

2009/11/30 by Leo Goldmakher, Goldmakher, Leo · 2 citations
Mathematics · #11L07 #11L40 #Analytic Number Theory Research #Analytic and geometric function theory #FOS: Mathematics #Finite Group Theory Research #Limits and Structures in Graph Theory #Mathematics and Applications #Number Theory (math.NT) #math.NT #msc:11L07 #msc:11L40

paper · pdf · doi:10.48550/arxiv.0911.5547

45 pages. Theorem 3 (and its proof in Section 9) added to this version

openalex publication_date 2009/11/30 · arxiv created 2010/10/21 · arxiv updated 2010/10/25 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

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.

Citations

• Exponential sums with multiplicative coefficients

Cited by

• Burgess-type character sum estimates over generalized arithmetic progressions of rank 2
• Sharp bounds for maximal sums of odd order Dirichlet characters

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