Fractional parts of polynomials over the primes

TL;DR

This paper establishes new inequalities for the fractional parts of polynomial values at prime numbers, improving previous bounds and extending understanding of their distribution.

Contribution

It introduces improved inequalities for the fractional parts of polynomials evaluated at primes, advancing prior results by Harman and Wong.

Findings

Inequalities hold for infinitely many primes.

Enhanced bounds over previous work.

Contributes to understanding polynomial fractional parts at primes.

Abstract

Let f be a polynomial with irrational leading coefficient. We obtain inequalities for the distance from the nearest integer of f(p) that hold for infinitely many primes p. These results improve work of Harman in 1981 and 1983 and Wong in 1997.