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- W2119550093 abstract "Let 8p(n)= 1 or 0 according as p|n or not. Since the functions E,p(n)i/p 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. 1. Determination of the approximating additive functions. Let 8p(n)= 1 or 0 according as pin or not, where p is a given prime number. Evidently N (1) N-' E 2,(n) = N-'[N/p] = 1 /p + 0(1/N), n=1 where [y] signifies the integer part of y. Thus the mean value of 8 (n) l/p is asymptotically zero. Since for p $ q, N N-L 2, {ep(n) l/p}{8q(n) l/q} (2 n=1 N[pq_ Npq_ _qp] pq (N the sequence {8 (n) l/p} is quasi-orthogonal asp runs through the prime numbers. This suggests that a good approximation to an arbitrary arithmetical functionf(n) can be found in the form (3) FN(f, n) = cl(f) + :_c/f ) p9(n), P:5< where 9p9(n)=a(p){8,,(n)-1/p}, suitably normalized to achieve N (4) N-'E 992(n) 1 (N -*+ o)) n=1 Presented in part to the Society, January 25, 1972 under the title Additive functions as quasi-orthogonal series for arithmeticalfunctions; received by the editors July 6, 1972. AMS (MOS) subject classifications (1970). Primary 1OK20; Secondary lOL10, IOA20, 1OH25." @default.
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- W2119550093 date "1973-01-01" @default.
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- W2119550093 title "Approximations of arithmetical functions by additive ones" @default.
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- W2119550093 doi "https://doi.org/10.1090/s0002-9939-1973-0311614-5" @default.
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