A Prime-Divisor Function

J. Knopfmacher
1973 Proceedings of the American Mathematical Society  
This note studies the asymptotic mean values over arithmetical progressions, the general distribution of values, and the maximum order of magnitude, of a certain natural prime-divisor function of positive integers. Consider the multiplicative arithmetical function ß defined by /S(l) = l and ß(n) = axa2 ■ ■ ■ ocr if n=p\lp%% ■ ■ ■ plr (pt prime, oct>0). Kendall and Rankin [2, p. 199] pointed out that this function has the finite mean value y 1 V * * £(2)£(3) , 0", lim -> ß(n) =-= 1.943 • • •
more » ... =-= 1.943 • • • Strangely, perhaps, there appears to be virtually no other information available about this natural arithmetical function. (See note added in proof.) This note makes a more detailed study of its asymptotic properties. 1. Average values and distribution.
doi:10.2307/2039376 fatcat:joyinbbanjeilj6bztqbpk3kj4