Summatory formula of the convolution of two arithmetical functions
The author proves a very general theorem (too complicated to be fully reproduced here) which furnishes the asymptotic formula for the summatory function of the convolution function \[ f\star g(n) = \sum_{d\mid n}f(d)g\left({n\over d}\right), \] provided sufficient information is available for the summatory functions of \(f(n), g(n)\) and their corresponding generating Dirichlet series. The proof starts by an application of the classical hyperbola method, and then proceeds by methods of complex integration. As an application, the author obtains two asymptotic formulas involving \(d(n) \equiv d_2(n)\), the number of divisors function. The first gives a sharp asymptotic formula for the summatory function of \(d(n)z^{\Omega(n)-\omega(n)}\), which is used to deduce the asymptotic formula for the summatory function of \(d(n)\) subject to the condition \( \Omega(n)-\omega(n) = q\), where \(\Omega(n)\) denotes the number of all prime factors of \(n\), \(\omega(n)\) denotes the number of distinct prime factors of \(n\), and \(z\) is a complex number such that \(|z|\leq R\) with \(R < 4/3\). The author remarks that the latter sum is a special case of the more general sum \[ \sum_{n\leq x,\;\Omega(n)-\omega(n) = q}d_k(n),\quad d_k(n) = \sum_{n_1n_2\cdots n_k=n}1, \] investigated by the reviewer [Publ. Inst. Math. 41, 31-41 (1987; Zbl 0648.10028)].
scientific article; zbMATH DE number 1798918