Title: Approximations of arithmetical functions by additive ones
Abstract: Let ev(n) = i or 0 according as/>|n or not.Since the functions Ev(ri)-\lp are quasi-orthogonal on the integers 1, 2, • • • , N with the relative frequency as measure, the theory of orthogonal expansions suggests an approximation of arbitrary arithmetical functions by strongly additive ones.In the present note, the approximating additive functions are determined and a sufficient condition is given for an arithmetical function to have an asymptotic distribution.Examples are given to illustrate the result. 'AT1 JV' qj NqlpJ pq Np the sequence {ep(n)-l/p} is quasi-orthogonal asp runs through the prime numbers.This suggests that a "good" approximation to an arbitrary arithmetical function f(n) can be found in the formwhere (pv(n)=a(p){ep(n)-\lp}, suitably normalized to achieve (4) N-1 J <fl(n) ~ 1 (N -+ oo)Presented in part to the Society, January 25, 1972 under the title Additive functions as quasi-orthogonal series for arithmetical functions; received by the editors July 6,1972.