Local theorems for integer-valued multiplicative functions with local conditions on powers of primes (Q1065858)

Let \(f\) be an integer-valued multiplicative function. Assuming the existence of a limiting distribution of the values \(f(p^ j)\) in the form \[ \#\{p\leq x,\;f(p^ j)=m\} = c_{mj} x^ a (\log^{-1}x + \text{smaller terms} + O(\log^{-2-\varepsilon}x)), \] a local limit law for \(f\) is found in the form \[ \#\{n\leq x,\;f(n)=m\} \sim cx^{\alpha} (\log x)^{\beta} (\log\log x)^{\gamma}; \] a slightly more general result is announced without proof.