A mean value theorem for binary digits

Alan Stein and Ivan Ernest Stux

Vol. 75 (1978), No. 2, 565–577

DOI: 10.2140/pjm.1978.75.565

Abstract

This paper continues the investigation of the dyadically additive function α defined by α(n) = the number of 1’s in the binary expansion of n.

Previously, Bellman and Shapiro (cf. “On a problem in additive number theory.” Annals of Mathematics, 49 (1948) 333–340) showed that ∑ _k=1^xα(k) ∼ xlog x∕2log 2. They then considered the iterates of α defined by α_q = α_q−1 ∘ α and observed that A_r(x) = ∑ _k=1^xα_r(k) is not asymptotic to any elementary function for r ≧ 2.

Mathematical Subject Classification

Primary: 10H25, 10H25

Secondary: 10A30

Milestones

Received: 14 September 1976

Revised: 16 May 1977

Published: 1 April 1978