Long nonnegative sums of Legendre symbols

TL;DR

This paper investigates the positivity of partial sums of Legendre symbols for primes, demonstrating that for certain rational and irrational fractions close to 1/3, these sums are nonnegative for most primes.

Contribution

It provides new results on the positivity of Legendre symbol sums for specific rational and irrational fractions, extending understanding of their distribution.

Findings

Positivity of $L(rac{1}{3}, p)$ for most primes.

Nonnegativity of $L(rac{ ext{rational}}{p})$ for certain fractions.

Empirical evidence supporting positivity in specified cases.

Abstract

For $0\leq \alpha<1$ 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 $|\alpha-\frac13|<2\cdot 10^{-6}$ and for rational $\alpha\leq \frac12$ with denominators in the set $\{1,2,3,4,5,6,8,12\}$ the inequality $L(\alpha,p)\geq 0$ holds for majority of primes.