Pólya–Vinogradov and the least quadratic nonresidue

Jonathan Bober, Leo Goldmakher

Research output: Contribution to journalArticle (Academic Journal)peer-review

5 Citations (Scopus)
296 Downloads (Pure)


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 (logk)1.4.
Original languageEnglish
Pages (from-to)853-863
Number of pages11
JournalMathematische Annalen
Issue number1
Early online date29 Dec 2015
Publication statusPublished - Oct 2016


Dive into the research topics of 'Pólya–Vinogradov and the least quadratic nonresidue'. Together they form a unique fingerprint.

Cite this