Vol. 63, No. 2, 1976

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A variance property for arithmetic functions

Joseph Eugene Collison

Vol. 63 (1976), No. 2, 347–355
Abstract

A pivotal point for certain problems in probabilistic number theory is that there exists a positive constant c such that for every member f of the family of additive complex valued arithmetic functions

∑
|f(m )− A(n)|2 ≦ cnD2 (n)
m≦n

where

        ∑
A (n) =    f(pα)p−α(1− p−1)
pα≦n

and

        ∑
D2(n) =     |f(pα)|2p−α(1− p−1),
pα≦n

pα being a power of a prime number. This paper considers the extension of this property in two directions suggested by Harold N. Shapiro. First, an investigation is made of when this property holds for weight functions other than w(m) 1. Second, it is shown that this property can be extended to various nonadditive arithmetic functions.

Mathematical Subject Classification
Primary: 10K20, 10K20
Milestones
Received: 26 August 1975
Published: 1 April 1976
Authors
Joseph Eugene Collison