Products in residue classes (Q1000628)

For coprime positive integers \(a\) and \(m\) let \(P(x;m,a)\) be the number of solutions to the congruence \(p_1p_2\equiv\pmod p\), in prime variables \(p_1\) and \(p_2\). One might expect that \(P(x;m,a)\) should be positive for \(x\gg_\varepsilon m^{1/2+\varepsilon}\). This was investigated by \textit{P. Erdős}, \textit{A. M. Odlyzko} and \textit{A. Sárközy} [Period. Math. Hung. 18, 229--239 (1987; Zbl 0625.10035)] assuming the generalized Riemann hypothesis, but they were unable to prove even that \(P(m;m,a) >0\). However they were able to provide a conditional treatment for \[ P(x,m):= \sum_{\alpha\leq m,(a,m)= 1}\Biggl(P(x;m,a)- {\pi(x)^2\over \phi(m)}\Biggr)^2 \] when \(x=m\). The present paper looks at unconditional results for the sums \[ R(x,M)= \sum_{M< m\leq 2M} P(x,m)\quad\text{and}\quad R_\pi(x,M)= \sum_{M< q\leq 2M} P(x,q), \] where \(q\) is restricted to primes. The trivial bounds for these have order \(x^4\) and \(x^4(\log x)^{-1}\) respectively, and the paper's first result improves these to give \[ R(x,M)\ll x^4(\log x)^{-1}+ Mx^2 \] and \[ R_\pi(x, M)\ll (M^{-1} x^4+ Mx^2)(\log x)^{-2} \] for any fixed \(A> 0\). The next theorem looks at the analogue of \(P(x;m,a)\) in which one replaces \(p_1\) and \(p_2\) by square-free integers. This is distinctly easier, and an asymptotic formula is achieved unconditionally as soon as \(x\gg m^{3/4+ \varepsilon}\). Other variants are also considered. For example if one looks at \(ps\equiv a\pmod m\) with \(p\) prime and \(s\) square-free, then one gets a non-trivial bound for the analogue of \(P(x,m)\) as soon as \(x\gg m^{1/2+\varepsilon}\).