Pólya-Vinogradov and the least quadratic nonresidue

Abstract

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 a new approach to nonresidues. The goal of this note is to demonstrate a new line of attack via long character sums, a currently 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 $(\log k{)}^{1.4}$.

9 pages; a few small corrections from the previous version

References in corpus (1)

• Averages of character sums

Cited by in corpus (4)

• The frequency and the structure of large character sums
• Least Prime Primitive Roots
• Consecutive Quadratic Residues And Quadratic Nonresidue Modulo $p$
• Note On The Maximal Primes Gaps