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 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 .

Three conjectures about character sums

“…One unlikely but currently plausible 'obstruction' to establishing this unconditionally is the possibility that χ(p) = 1 for all primes p q c , in which case c 1 , c 2 , c 3 ≫ c, or indeed if χ is 1-pretentious for the primes up to q. 1 Inspired by connections highlighted in [2,7,20] we show that improving any one of these bounds will, more-or-less, improve the others.…”

“…Following an idea from [2], Proposition 3 has the following consequence for quadratic non-residues: 2 Corollary 2. Fix ∆ ∈ ( 2 π , 1).…”

“…Proof of Theorem 3. Let c 0 := 1 log X , A = 2 and N and T be as in Lemma 4 so that T (log X) 2 . By a quantitative form of Perron's formula [24, Cor.…”

A Refinement of the Burgess Bound for Character Sums

“…We expect this to be true for smaller values of N although this problem is much less understood. In the special case M = 0, the Generalized Riemann Hypothesis (GRH) implies that (1.3) 0<n N χ(n) N 1/2 q o (1) , which is nontrivial provided N q ε and is essentially optimal. Although the conditional bound (1.3) on the GRH is well-known, see for example [18,Section 1]; it may not be easy to find a direct reference, however it can be easily derived from [12,Theorem 2].…”

Improving the Burgess bound via Pólya-Vinogradov

“…It is a fun exercise to construct a family of completely multiplicative functions f : Z → [−1, 1] for which (1.1) fails. 1 For our application, however, we will only need to compare the two means in the situation that both x and |M f (x)| are large, and in this case we will show that (1.1) does hold. More precisely: It would be interesting to find a more explicit relationship between the two means which holds even when they are small.…”