[Paper Review] Pólya-Vinogradov and the least quadratic nonresidue

Jonathan Bober, Leo Goldmakher|arXiv (Cornell University)|Nov 29, 2013

Analytic Number Theory Research5 references4 citations

TL;DR

This paper establishes a novel connection between long character sum estimates and bounds on the least quadratic nonresidue. By showing that improvements to the Pólya–Vinogradov inequality for even characters directly imply stronger bounds on the least nonresidue for odd characters, it opens a new path to improving Burgess's classical exponent, with conditional results yielding $ n_p \ll (\log p)^{1.4} $ under a conjecture on long character sums.

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 $(\log k)^{1.4}$.

Motivation & Objective

• To establish a new connection between long character sum estimates and bounds on the least quadratic nonresidue.
• To show that improvements in the Pólya–Vinogradov inequality for even characters lead to stronger bounds on the least nonresidue for odd characters.
• To explore the implications of recent advances in long character sum theory for the longstanding problem of bounding the least quadratic nonresidue.
• To investigate the conditional bound $ n_\xi \ll (\log k)^{1.4} $ under a conjecture on long character sums.
• To examine the limitations of the method and its relationship to the size of $ L(1,\xi) $ and the Generalized Riemann Hypothesis.

Proposed method

• The paper uses the Pólya–Vinogradov inequality as a starting point, relating the maximum partial sum $ M(\chi) $ of a character $ \chi $ to bounds on the least nonresidue $ n_\xi $ of an odd character $ \xi $.
• It constructs an even character $ \chi \mod 3k $ from an odd character $ \xi \mod k $, leveraging the sum $ \sum_{n \leq y^\alpha} \frac{\xi(n)}{n} $ to derive lower bounds on $ M(\chi) $.
• The method applies the Dickman–de Bruijn function $ \rho(u) $ to model the distribution of smooth integers and estimate partial sums of $ \xi(n)/n $, assuming $ \xi(n) = 1 $ on $ y $-smooth numbers.
• It uses the identity $ \left| \sum_{n \leq y^\alpha} \frac{\xi(n)}{n} \right| \geq \sum_{n \leq y^\alpha} \frac{1}{n} - 2 \sum_{P^+(n) > y} \frac{1}{n} $ to bound the sum from below, relying on coprimality and smoothness conditions.
• The key inequality $ M(\chi) \gtrsim \frac{\sqrt{k}}{\pi} (-2\alpha \log \alpha + 3\alpha - 2) \log y $ is maximized at $ \alpha = \sqrt{e} $, leading to the main result.
• Heuristic arguments based on pretentious character theory and the distribution of smooth numbers lead to the conjecture that $ \log n_\xi \lesssim \left( \frac{\pi}{e^\gamma} \right) \frac{M(\chi)}{\sqrt{k}} $, which implies $ n_\xi \ll (\log k)^{1+o(1)} $ under the conjectured bound on $ M(\chi) $.

Experimental results

Research questions

• RQ1 Can improvements in the Pólya–Vinogradov inequality for even characters lead to better bounds on the least quadratic nonresidue for odd characters?
• RQ2 What is the quantitative relationship between the maximum partial sum $ M(\chi) $ of an even character and the least nonresidue $ n_\xi $ of an odd character?
• RQ3 To what extent can the theory of long character sums, particularly under the Granville–Soundararajan conjecture, improve known bounds on the least nonresidue?
• RQ4 Is there a conditional bound on $ n_\xi $ that improves upon the Burgess exponent $ \frac{1}{4\sqrt{e}} $, and if so, what is its strength?
• RQ5 How does the size of $ L(1,\xi) $ relate to the least nonresidue and the maximum partial sum $ M(\chi) $ of an associated even character?

Key findings

• Improving the constant in the Pólya–Vinogradov inequality for even characters would directly lead to an improvement of the Burgess exponent $ \frac{1}{4\sqrt{e}} $ for the least quadratic nonresidue $ n_p $, particularly for primes $ p \equiv 3 \pmod{4} $.
• Under the conjecture that $ M(\chi) \leq \left( \frac{e^\gamma}{\pi\sqrt{3}} + o(1) \right) \sqrt{q} \log \log q $ for all even primitive characters $ \chi \mod q $, the paper shows $ n_\xi \ll (\log k)^{c+o(1)} $ with $ c \approx 1.37 $, improving on Ankeny's $ \ll (\log p)^2 $ bound under GRH.
• The bound $ n_\xi \ll (\log k)^{1.4} $ is achieved conditionally under the same conjecture, representing a significant strengthening of the state of the art.
• The paper demonstrates that if $ M(\chi) = o(\sqrt{q} \log q) $ for all even characters $ \chi \mod q $, then $ n_\xi \ll k^{o(1)} $, suggesting a path to sub-logarithmic bounds.
• The method shows that the maximum size of the least nonresidue is deeply tied to the maximum size of $ L(1,\xi) $, and that $ L(1,\xi) $ is likely large when $ n_\xi $ is large.
• The paper proves that for any primitive even character $ \chi \mod q $, either $ n_\chi \leq \exp((\log q)^{5/6+\epsilon}) $ or $ M(\chi) \leq \sqrt{q} (\log q)^{2/3+\epsilon} $, showing that at least one of the two quantities must be small.