Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  359.10038
Autor:  Alladi, K.; Erdös, Paul
Title:  On an additive arithmetic function. (In English)
Source:  Pac. J. Math. 71, 275-294 (1977).
Review:  Let n be a positive integer, n = prodi = 1rpi\alpha1 in canonical form, and let A(n) = sumi = 1r\alpha1p1. Clearly A is an additive arithmetic function. Assume the primes p1 are arranged so that p1 \leq p2 < ... < pr. Define P1(n) = pr and, in general, Pk(n) = P1(n/P1(n)... Pk-1(n)) for k \leq sumi = 1r\alpha1 and Pk(n) = 0 for k > sumi = 1r\alpha1. If f and g are arithmetic functions such that sumn \leq xf(n) ~ sumn < xg(n), and if g is a well behaved function (e.g. polynomial, exponential), then g is referred to as the average order of f. It is proved that for all positive integers, we have

sumn \leq xPm(n) ~ sumn \leq x{A(n)-P1(n)-... -Pm-1(n)} ~ kmx1+(1/m)/(log x)m

where km is a positive constant depending only on m. It follows almost immediately from this theorem that the average order of A(n) is \pi2n/6 log n. Let A^*(n) = sumi = 1rp1. Then the average order of A^*(n) is also \pi2n/6 log n, and the average order of A(n)-A^*(n) is log log n. For any fixed positive integer M, the set of solutions to A(n)-A^*(n) = M has a positive natural density. Now A(n) = n if and only if n is a prime or n = 4. Call n a ''special number'' if n\equiv O(mod A(n)) and n\neq A(n), and let {ln} be the sequence of special numbers. This paper's first author has previously proved that the sequence {ln} is infinite [Srinivasa Ramanujan Commemoration Volume, Oxford Press, Madras, India, (1974) part II] . Denote by L(x) the number of \elln \leq x. It is shown that there exist positive constants c, c' such that

L(x) = O(xe-c\sqrt{log x log log x}) and L(x) >> xe-c'\sqrt{log x log log x}.

Finally, let \alpha(n) = (-1)A(n). It is proved that there exists a positive constant c'' such that

sum1 \leq n \leq x\alpha(n) = O(xe-{c''\sqrt log x log log x}),

and that sumn = 1oo\alpha(n)/n = 0.
Reviewer:  B.Garrison
Classif.:  * 11N37 Asymptotic results on arithmetic functions
                   11K65 Arithmetic functions (probabilistic number theory)

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