Long nonnegative sums of Legendre symbols

Abstract

For $0\le \alpha <1$ and prime number $p$ let $L(\alpha ,p)$ be the sum of the first $[\alpha p]$ values of Legendre symbol modulo $p$. We study positivity of $L(\alpha ,p)$ and prove that for $|\alpha -\frac{1}{3}|<2\cdot 10^{-6}$ and for rational $\alpha \le \frac{1}{2}$ with denominators in the set $\{1,2,3,4,5,6,8,12\}$ the inequality $L(\alpha ,p)\ge 0$ holds for majority of primes.

Typos corrected