A localized Erdős-Wintner theorem
For \(1\leq y\leq x\), let \(F(x,y,z)\) denote the frequency of those integers \(n\) in the interval \(x-y<n\leq x\) for which the additive arithmetic function \(f\) satisfies \(f(n)\leq z\). A well-known theorem of Erdős [\textit{P. Erdős} and \textit{A. Wintner}, Am. J. Math. 61, 713--721 (1939; Zbl 0022.00903)] asserts that \(F(x,x,z)\) converges weakly to a distribution as \(x\to \infty\) if and only if the three series \(\sum 1/p\), \(\sum f(p)/p\), \(\sum f(p)^2/p\) taken over the prime numbers converge. Improving a result of \textit{A. Hildebrand} [Can. J. Math. 39, 646--672 (1987; Zbl 0618.10040)] it is shown that the same condition is necessary and sufficient for the weak convergence, as \(j\to \infty\), of the frequencies \(F(N_j,M_j,z)\), provided \(1\leq M_j\leq N_j\), \(N_j\to \infty\), \((\log M_j)/\log N_j\to 1\), and there is a constant \(c>1\) so that \(N_{j+1}\leq N_j^c\) for all \(j\). The proof is quite different from the classical one of Erdős and Wintner.
scientific article; zbMATH DE number 4027598