Mean square of an error term related to a certain exponential sum involving the divisor function (Q2707562)

As a generalization of the error term \(\Delta (x)\) in Dirichlet's divisor problem, let \(\Delta (x;b/a)\) denote the analogous error term in the asymptotic formula for the sum function of \(d(n)e(na/b)\) for coprime integers \(a\) and \(b\) with \(a\geq 1\). The author is interested in the error term \(F(x;b/a)\) in the mean square formula for \(\Delta (x;b/a)\). Pointwise, mean value and omega estimates for the correponding function \(F(x)=F(x;1)\) related to the ordinary divisor problem have been obtained by Preissman, Lau and Tsang, and the author generalizes these results. For instance, \(F(x,b/a) \ll a^2x\log ^4 x +a^{4+\varepsilon }\) for \(a\leq x\), and \(F(x;b/a)=\Omega (x \log ^2x)\) if \(a\) is fixed and \(x\) tend to infinity. The proofs are based on formulae of the Voronoi type for \(\Delta (x;b/a)\) and estimates for the error term in the additive divisor problem.NEWLINENEWLINEFor the entire collection see [Zbl 0932.00040].