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

Abstract

We show that there is a positive constant $c_0$ such that \[\sum_{n\le x}μ^2(n^2+1)c_0x+O_{\varepsilon}(x^{7/12+\varepsilon})\] for any fixed $\epsilon >0$. 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."

A small correction has been made in the final section

Cited by in corpus (2)

• Pairs of k-free Numbers, consecutive square-full Numbers
• Errata and Addenda to Mathematical Constants