Sur une inégalité de la théorie probabiliste des nombres. (On an inequality in probabilistic number theory)
Let \(\Lambda\) be a semigroup with norm N( ), generated by a set of ''primes'' p. The author shows that, under certain hypotheses on \(\Lambda\), the following analogue of the well-known Turán-Kubilius inequality holds: For any additive function \(f: \Lambda\to {\mathbb{C}}\) and every \(x\geq 1\), \[ (1/x)\sum_{N(a)\leq x}| f(a)-\sum_{N(p^ m)\leq x}f(p^ m)/N(p^ m)|^ 2\leq C\sum_{N(p^ m)\leq x}| f(p^ m)|^ 2/N(p^ m)\quad, \] where C is an absolute constant. Reviewer's remark: Essentially the same inequality, though under somewhat different assumptions on \(\Lambda\), is given in the paper [\textit{Z. Juškis} (Z. Yushkis), Limit theorems for additive functions defined on ordered semigroups with regular norm (Russian), Litov. Mat. Sb. 4, 565-603 (1964; Zbl 0151.032)], of which the author apparently was unaware.
scientific article; zbMATH DE number 4006363