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Large character sums: Pretentious characters and the Polya-Vinogradov Theorem

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Andrew Granville, K. Soundararajan

Number Theory · Submitted 6 Mar 2005

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim One of the main results of this paper provides a substantial improvement of the Polya-Vinogradov bound for characters of odd, bounded order.

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arXiv IDmath/0503113 (version 1)
Pith ID52MAEILJ, Pith's permanent record of this paper and its review
CitationAndrew Granville, K. Soundararajan. "Large character sums: Pretentious characters and the Polya-Vinogradov Theorem." arXiv:math/0503113 (2005). https://pith.science/paper/52MAEILJ
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,227 input tokens.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.13 · human participants 0.01 · controlled comparison 0.07 · randomized assignment 0.01 · states quantitative result 0.31 · compares with prior results 0.96 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

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 GRH. 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 finds a hidden structure among these characters.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 In 1918 Polya and Vinogradov gave an upper bound for the maximal size of character sums which still remains the best known general estimate. background 1.00 0.02 0.04 asserted 1.00
s1 One of the main results of this paper provides a substantial improvement of the Polya-Vinogradov bound for characters of odd, bounded order. result 0.80 0.94 0.81 asserted 1.00
s2 In 1977 Montgomery and Vaughan showed how the Polya-Vinogradov inequality may be sharpened assuming the GRH. background 1.00 0.03 0.04 asserted 0.75
s3 We give a simple proof of their estimate, and provide an improvement for characters of odd, bounded order. contribution 0.96 0.96 0.14 asserted 1.00
s4 The paper also gives characterizations of the characters for which the maximal character sum is large, and finds a hidden structure among these characters. result 0.72 0.96 0.14 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

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 GRH. 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 finds a hidden structure among these characters.

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