The Ingham divisor problem on the set of numbers without \(k\)th powers (Q5951056)

This paper proves an asymptotic formula for \[ \mathop{{\sum}^*}_{n\leq x} d(n)d(n+1), \] where \(\sum^*\) indicates that \(n\) is \(k\)th power free, and \(n+1\) is \(l\)th power free. The main term takes the form \(xQ_{k,l}(\log x)\), where \(Q_{k,l}\) is a quadratic polynomial, while the error term is \(O_{k,l} (x^{5/6+1/(6h)+ \varepsilon})\) for any \(\varepsilon> 0\), with \(h= \min(k,l)\). The proof uses an estimate for \[ \sum_{n\leq x, n\equiv a\pmod q}d(n), \] based ultimately on Weil's bound for the Kloosterman sum.