Large character sums: Pretentious characters and the Pólya-Vinogradov theorem

By Andrew Granville, Kannan Soundararajan. 2005

171 citations recorded by the source corpus.

In 1918 Polya and Vinogradov gave an upper bound for the maximal size of character sums, which still remains the best known general estimate. One of the main results of this paper provides a substantial improvement of the Polya-Vinogradov bound for characters of odd, bounded order. In 1977 Montgomery and Vaughan showed how the Polya-Vinogradov inequality may be sharpened assuming the Generalized Riemann Hypothesis. We give a simple proof of their estimate and provide an improvement for characters of odd, bounded order. The paper also gives characterizations of the characters for which the maximal character sum is large, and it finds a hidden structure among these characters

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/a6abff8f0ae6f65075fcb6923518be1f2456fdbf

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In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

Citations in the corpus

Higher order Fourier analysis of multiplicative functions and applications (2014, background): “Lending terminology from [18], condition (iv) states that a multiplicative function is aperiodic unless it “pretends” to be χq(n)n it for some Dirichlet character χq and some t ∈ R.”
Weak subconvexity for central values of $L$-functions (2008, background): “The next Lemma is inspired by the ideas in [9] and [10].”