Quadratic characters with positive partial sums

TL;DR

This paper investigates primes for which quadratic character sums are non-negative over all initial segments, providing an upper bound on their density within the primes.

Contribution

It establishes a new upper bound on the count of such primes, refining understanding of quadratic character sum behavior over primes.

Findings

Upper bound on the number of primes with non-negative quadratic character sums

The bound involves a specific logarithmic decay with a constant c≈0.0368

Results contribute to understanding the distribution of primes with special quadratic properties

Abstract

Let $\mathcal L^+$ be the set of all primes $p$ for which the sums of $\left(\frac{n}{p}\right)$ over the interval $[1,N]$ are non-negative for all $N$. We prove that the estimate \[ |\mathcal L^+\cap [1,x]|\ll \frac{x}{\ln x(\ln\ln x)^{c-o(1)}} \] holds for $c\approx 0.0368$