Sieving for mass equidistribution (Q601877)

Let \(\tau_m(n)\) be the number of ways to represent \(n\) as product of \(m\) natural numbers. Let \(\lambda_1(n)\) and \(\lambda_2(n)\) be multiplicative functions which satisfy \(|\lambda_i(n)|\leq \tau_m(n)\) for some \(m\). The author proves: for any \(0<\varepsilon<1\), any \(x\) sufficiently large with respect to \(\varepsilon\) and any fixed integer \(0<|l|\leq x\) we have \[ \sum_{n\leq x}|\lambda_1(n)\lambda_2(n+l)|\ll x(\log x)^\varepsilon M(x)\tau(|l|), \] where \[ M(x):=\frac{1}{(\log x)^2}\prod_{p\leq \exp(\frac{\log x}{\varepsilon\log\log x})}\left(1+\frac{|\lambda_1(p)|}{p}\right)\left(1+\frac{|\lambda_2(p)|}{p}\right). \] He applies this result to approach the mass equidistribution conjecture.