Sur certaines équations fonctionnelles arithmétiques. (On certain arithmetical functional equations)
The motivation for this extensive study of certain classes of additive and multiplicative arithmetical functions is the additive function, called by the authors the Gutman-Ivić-Matula function [see \textit{I. Gutman} and the reviewer, Discrete Math. 150, 131-142 (1996; Zbl 0854.05035)]. It is the unique completely additive function satisfying \(f(p_k) = 1 + f(k),\) where \(p_k\) is the \(k\)-th prime. The authors consider more generally \(\mathcal E\), the class of completely additive functions which satisfy \[ F(p_k) = F(k) + b\log\log k + a + O\left({\log\log k\over\log k}\right) \] for some \(a,b\in\mathbb R\). Théorème 1.1 evaluates the mean value of any arithmetical function \(g(n)\) satisfying \(g(n) \ll \log^\alpha n (\alpha > 0)\) in terms of \[ R(x,g) := {1\over x}\sum_{n\leq x}g(n) - {1\over x}\sum_{p_k\leq x}g(k)\left[{x\over p_k}\right]. \] The method of proof is based on the use of certain approximate functional equations of independent interest. In Théorème 1.2 they obtain the asymptotic evaluation of moments of functions from \(\mathcal E\), more precisely for \(f_0 := f - \Phi_1(f)\log\), where \(f\in {\mathcal E}\) and \(\Phi_1(f)\) is a suitable constant depending on \(f\). From this result they deduce the following one. Let \(f\) be a completely additive arithmetical non-zero function satisfying \(f(p_k) \geq f(k)\) for \(k\in \mathbb N\). Then \[ \sum_{n\leq x}f(n)\gg x\log x, \] which implies that the constant \(\Phi_1(f)\) is strictly positive. Also if \(f\in {\mathcal E}\) and the second moment of \(f_0\) (defined above) is \(o(x\log x)\) as \(x\to\infty\), then \(f(n) = \Phi_1(f)\log n\). The remaining two theorems of the paper deal with asymptotic results on certain multiplicative functions of at most unit modulus. They are used to yield information on the distribution of values of functions from \({\mathcal E}\). For example, if \[ h_N(n;\tau) = \exp(i\tau f_0(n)/\sqrt{\Phi_2\log N}) \quad(\tau\in\mathbb R, N\geq 2), \] where \(\Phi_2\) is a suitable constant, then we have uniformly \[ {1\over N}\sum_{n\leq N}h_N(n;\tau) = \text{e}^{-{1\over 2}\tau^2} + O\left((\tau + \tau^3){\log\log N\over\sqrt{\log N}}\right). \] The proofs of the authors' results show a veritable display of various methods from analytic and probabilistic number theory, including many lemmas of intrinsic interest. This is a remarkable paper, with a wealth of results, whose study will reward the reader.
scientific article; zbMATH DE number 1528712