Vol. 97, No. 2, 1981

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ISSN: 0030-8730
An analogue of Kolmogorov’s inequality for a class of additive arithmetic functions

Joseph Eugene Collison

Vol. 97 (1981), No. 2, 319–325
Abstract

Let f be a complex valued additive number theoretic function (i.e., f(mn) = f(m) + f(n) if m and n are relatively prime). This paper shows that D2(pα)pα = O(D2(n)) or |f(pα)|pα = O(D(n)) (where the summations are over those pα n, pα being a prime raised to a power) is sufficient to guarantee that the following analogue of Kolmogorov’s inequality holds:

νn{Max|fk(m)− A (k )| > tD (n )} = O(t−2)
k≦n

where, if pαm denotes the fact that pα divides m but pα+1 does not (i.e., pα exactly divides m), then

A(n) = pαnf(pα)pα,
D2(n) = pαn|f(pα)|2pα,
fk(m) = α
ppα≦∥kmf(pα),
and
           ∑
νn(𝒮) = n−1   1
m≦n
m∈𝒮

for any set 𝒮.

Mathematical Subject Classification
Primary: 10H25, 10H25
Secondary: 10K20
Milestones
Received: 22 July 1980
Revised: 24 October 1980
Published: 1 December 1981
Authors
Joseph Eugene Collison