Estimates on Lattice Points in the Circle

TL;DR

This paper derives new estimates for the difference between lattice point counts and circle area, improving bounds with specific exponents, notably achieving a new estimate with b=5/16 and the best at approximately 0.3091.

Contribution

It introduces improved bounds on lattice point estimates in circles, including a novel estimate with b=5/16 and the best known bound at about 0.3091.

Findings

New estimate with b=5/16

Best estimate at approximately 0.3091

Estimates are of the form O(log R * R^{2b})

Abstract

This paper provides estimates on the difference between the number of integer lattice points an a circle centered at the origin and the area. The estimates have the form "Big O" of the product of logarithm of the radius and the radius raised to a power 2b. The simplest new estimate obtained is b = 5/16; the best is b = 1507/4875 = .309128205...