Additive functions with asymptotically finite supports (Q1415513)

Let \(f(x,n)\) be a strongly additive function on the integers \(n= 1,2,\dots,x\) for a fixed integer \(x\), where \(f(x,p)\) is either \(0\) or \(1\) for each prime not exceeding \(x\), and \(f(x,p)\) may change both with \(x\) and \(p\). If there is a constant \(c\) such that the relative frequency of the integers \(n\), not exceeding \(x\), such that \(f(x,n)> c\), converges to \(0\), \(f(x,n)\) is said to have a finite support. The author gives a simple characterization of \(f(x,n)\) to have a finite support. The major tool is to establish inequalities on the binomial moments of \(f(x,n)\).