Square-free values of $n^2+1$

TL;DR

This paper improves the error term in counting square-free values of n^2+1, using the determinant method to analyze rational points near algebraic curves, advancing previous results from 1931.

Contribution

It provides a sharper error estimate for the distribution of square-free values of n^2+1, improving the classical result by Estermann.

Findings

Error term improved from 2/3 to 7/12+ε

Uses determinant method for counting rational points

Establishes positive constant c_0 for the asymptotic formula

Abstract

We show that there is a positive constant $c_0$ such that \[\sum_{n\le x}\mu^2(n^2+1)c_0x+O_{\varepsilon}(x^{7/12+\varepsilon})\] for any fixed $\varepsilon>0$. This improves a result of Estermann [3] from 1931, in which the error term had an exponent 2/3. The proof involves counting rational points near an algebraic curve, which is done via the "determinant method."