Every integer can be written as a square plus a squarefree (Q2674686)

In this paper, the author shows that every integer can be written as the sum of two integers, one perfect square and one squarefree. Moreover, let \(r(n)\) count the number of representations of an integer \(n\) as a sum of a square plus a squarefree, the author establishes the following asymptotic formula \[ r(n)=n^{\frac{1}{2}}\prod_p \left( 1-\frac{g(p)}{p^2} \right)+E(n), \] where \[ E(n)<\frac{24}{(\log n)^\frac{3}{4}}n^{\frac{\log 2+{0.342}}{\log\log n}}, \] and \(g\) is the completely multiplicative function which, at prime \(p\), counts the number of solutions to the congruence class \(x^2 \equiv n \pmod {p^2}\). If one considers the number \(r^\ast(n)\) of primitive representations of \(n\) as a square plus squarefree, i.e., representations in which both the square and the squarefree are coprime with \(n\), the author also establishes a similar asymptotic formula of \(r^\ast(n)\) to that of \(r(n)\). The above result is deeply related with the divisor function. The author proves the following upper bound for the divisor function fully explicit. For every \(\varepsilon>0\), and every \(n>e^{e^{\frac{14}{\sqrt{1+28\varepsilon}-1}}}\) we have \[ \tau(n)<n^{\frac{\log2+\varepsilon}{\log\log n}}. \] In particular, for any \(n\geq 3\) \[ \tau(n)<n^{\frac{3}{\log\log n}}. \]