Pólya–Vinogradov and the least quadratic nonresidue

It is well-known that cancellation in short character sums (e.g. Burgess’ estimates) yields bounds on the least quadratic nonresidue. Scant progress has been made on short character sums since Burgess’ work, so it is desirable to find another approach to nonresidues. In this note we formulate a new...

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Bibliographic Details
Published in:Mathematische annalen 2016-10, Vol.366 (1-2), p.853-863
Main Authors: Bober, Jonathan W., Goldmakher, Leo
Format: Article
Language:English
Subjects:
Mathematics
Mathematics and Statistics
Sums
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Summary:It is well-known that cancellation in short character sums (e.g. Burgess’ estimates) yields bounds on the least quadratic nonresidue. Scant progress has been made on short character sums since Burgess’ work, so it is desirable to find another approach to nonresidues. In this note we formulate a new line of attack on the least nonresidue via long character sums, an active area of research. Among other results, we demonstrate that improving the constant in the Pólya–Vinogradov inequality would lead to significant progress on nonresidues. Moreover, conditionally on a conjecture on long character sums, we show that the least nonresidue for any odd primitive character (mod k ) is bounded by ( log k ) 1.4 .
ISSN:0025-5831
1432-1807
DOI:10.1007/s00208-015-1353-2