A quadratic divisor problem
Let \[ D_ f(a,b;h) = \sum_{am \pm bn = h} \tau(m)\tau(n) f(am,bn), \] where \(\tau(n)\) is the usual divisor function, \(a\), \(b\), and \(h\) are parameters, and \(f\) is a smooth function. A general formula with an explicit main term and with an estimate for the error term is obtained for this sum under suitable conditions on the function \(f\), for both choices of \(\pm\). In particular, the case \(a = b = 1\) is related to the classical additive divisor problem. To give a sample of the results, let us consider the sum \(\sum_{m \leq M}\tau (m) \tau(am + h)\) for \(a,h,M \geq 1\). The error term in the asymptotic formula for this sum is \(O(a^{1/9}(aM + h)^{2/9} M^{2/3+\varepsilon})\). For \(a = 1\), this is weaker than the known error term in the additive divisor problem. However, the point of the results lie in their generality. A key role in the argument is played by a formula for the indicator function of zero among all integers; this ``\(\delta\)-method'' can be viewed as a discrete analog of the circle method. The deepest result needed is Weil's bound for Kloosterman sums. Unlike most of the recent work on the additive divisor problem, the present approach does not invoke the spectral theory of automorphic functions.
scientific article; zbMATH DE number 549843