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

D. R. Heath-Brown

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

Number Theory 2012-05-10 v3

Authors: D. R. Heath-Brown

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."

Keywords

number theory multiplicative functions divisor function

Cite

@article{arxiv.1010.6217,
  title  = {Square-free values of $n^2+1$},
  author = {D. R. Heath-Brown},
  journal= {arXiv preprint arXiv:1010.6217},
  year   = {2012}
}

Comments

A small correction has been made in the final section

Related papers

View all related →