Square-free values of \(n^2+1\) (Q2919662)

Let \(N(x)\) denote the number of positive integers \(n \leq x\) such that \(n^2+1\) is square-free. \textit{T. Estermann} [Math. Ann. 105, 653--662 (1931; Zbl 0003.15001)] proved that NEWLINE\[NEWLINE N(x) = c_0x + O(x^{2/3}\log x), NEWLINE\]NEWLINE with \(c_0 > 0\) an explicit arithmetic constant, defined by an Euler product. In the paper under review, the author proves that the error term in the above asymptotic formula can be replaced by \(O_{\varepsilon}(x^{7/12+\varepsilon})\) for any fixed \(\varepsilon> 0\). The key ingredient in the proof are sharp upper bounds for the number of solutions \((a,b,n)\) of the equation \(a^2b = n^2 + 1\) with \(A < a \leq 2A\) and \(B < b \leq 2B\). The author obtains the necessary estimates using the ``determinant method'' from his earlier work on density of rational points on curves and surfaces [J. Number Theory 129, No. 6, 1579--1594 (2009; Zbl 1182.11017)]. He also outlines a small further improvement on the main result, in which the exponent \(7/12 = 0.58\overline 3\) is reduced to \(46/81 = 0.\overline{567901234}\).