Vol. 77, No. 2, 1978

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Central moments for arithmetic functions

Joseph Eugene Collison

Vol. 77 (1978), No. 2, 307–314
Abstract

The only central moment considered in probabilistic number theory up until now has been the “variance” of an arithmetic function. This paper considers the case of higher central moments for such functions. It will be shown that if f is an additive complex valued arithmetic function then

 ∑                                   ∑
|f(m)− A (n)|2K = O(n(log logn )2K− 2    |f (pα)|2Kp −α)
m ≦n                                 pα≦n

where K is a positive integer and

A(n) = ∑   f(pα)p− α.
pα≦n

It will also be shown that if f is an additive real valued arithmetic function and K is an odd positive integer, then

 ∑                                      ∑
(f (m )− A(n))K = O(n(log log n)K −2+1∕K     |f(pα)|Kp −α).
m ≦n                                   pα≦n

Mathematical Subject Classification
Primary: 10K20, 10K20
Milestones
Received: 19 May 1977
Published: 1 August 1978
Authors
Joseph Eugene Collison