The local behaviour of some additive functions (Q1324780)

Let \(f\) be an integer-valued additive arithmetic function, and let \[ N_ x(a)= N_ x(a,f)= \#\{n\leq x:\;f(n)=a\}. \] In the case \(f(n)= \omega(n)\) a classical result of Sathe and Selberg gives an asymptotic formula for \(N_ x(a)\) that holds uniformly in the range \(a\leq C\log\log x\), where \(C\) is any given constant. The author of the present paper obtains asymptotic formulae with similar uniformity for functions \(f\) that satisfy, aside from some minor hypotheses on the values \(f(p^ k)\) for \(k\geq 2\) and on large values of \(f(p)\), an estimate of the type \[ \#\{p\leq x:\;f(p)=a\}= \pi(x) \Biggl(\lambda_ a+ O\biggl( {1\over {(\log x)^ \alpha}} \biggr) \Biggr) \] with a sufficiently large constant \(\alpha\) and suitable constants \(\lambda_ a\). The proof depends on a quantitative mean value theorem for multiplicative functions, due to \textit{R. Skrabutenas} [Lit. Mat. Sb. 14, No. 2, 115-125 (1974; Zbl 0302.10038)].