Mean value theorems for binary Egyptian fractions. II (Q2919688)

Let \(a\) be a fixed positive integer and denote by \(R(n;a)\) the number of positive integer solutions to the Diophantine equation NEWLINE\[NEWLINE {a\over n}={1\over x}+{1\over y}. NEWLINE\]NEWLINE In a previous paper [J. Number Theory 131, No. 9, 1641--1656 (2011; Zbl 1239.11038)], the authors have studied the mean value \(S(N;a)=\sum_{{n\leq N\atop (n,a)=1}} R(n,a)\).NEWLINENEWLINELet \(\chi\) denote the typical character modulo \(a\). Then, as remarked in the paper, NEWLINE\[NEWLINE R(n;a)={1\over\varphi(a)}\sum_{\chi\bmod\;a}\overline{\chi}(-n)\sum_{u|n^2}\chi(u).NEWLINE\]NEWLINE The main result of the paper is the following estimation: NEWLINE\[NEWLINE \sum_{{n\leq N\atop (n,a)=1}} \left|R(n;a) - {1\over\varphi(a)}\sum_{{\chi\bmod a\atop \chi^2=\chi_0}}\overline{\chi}(-n)\sum_{u|n^2}\chi(u) \right|^2 \ll_a N\log^2 N, NEWLINE\]NEWLINE where \(\chi_0\) is the principal character modulo \(a\), and the subscript \(a\) in \(\ll_a\) indicates that the implicit constant depends, at most, on \(a\).NEWLINENEWLINEAs a consequence of this result, the authors prove that (for \(a\) fixed) the function \(f(n)=(\log 3)\log\log n\) is a normal order of the function \(R(n;a)\).