Mean behaviour and distribution properties of multiplicative functions
Let \(f: \mathbb{N}\to\mathbb{C}\) be a multiplicative function and \({\mathfrak a}= (a_n)\) a sequence of natural numbers. The function \(f\) has mean-value \(m(f,{\mathfrak a})\) on \({\mathfrak a}\), if the normalized summatory function \(M(f,{\mathfrak a}, x):= {1\over x}\sum_{n\leq x} f(a_n)\) converges to \(M(f,{\mathfrak a})\) as \(x\to\infty\). The authors give criteria for existence of \(M(f,{\mathfrak a})\) and estimates for \(M(f,{\mathfrak a}, x)- M(f,{\mathfrak a})\). They discuss examples as \(a_n= n\), \(a_n= 2^n- 1\) (Mersenne numbers), \(a_n= p_n- 1\) (\(n\)th shifted prime) and visualize their results. Finally, they prove distribution properties and cluster points of the image sequence \(f({\mathfrak a})\).
scientific article; zbMATH DE number 2153535
scientific article