Prime lattice points in ovals

TL;DR

This paper investigates the distribution of prime lattice points within dilated convex ovals, providing asymptotic formulas and bounds for the remainder term under the Riemann Hypothesis and additional assumptions on zeta zeros.

Contribution

It offers new bounds and a distribution formula for the prime lattice point counting function in convex domains, assuming RH and zero independence.

Findings

Sharp upper bounds for the remainder term under RH

Asymptotic formula for prime lattice points in convex ovals

Distribution function for normalized remainder term

Abstract

We study the distribution of lattice points with prime coordinates lying in the dilate of a convex planar domain having smooth boundary, with nowhere vanishing curvature. Counting lattice points weighted by a von Mangoldt function gives an asymptotic formula, with the main term being the area of the dilated domain, and our goal is to study the remainder term. Assuming the Riemann Hypothesis, we give a sharp upper bound, and further assuming that the positive imaginary parts of the zeros of the Riemann zeta functions are linearly independent over the rationals allows us to give a formula for the value distribution function of the properly normalized remainder term.