Vol. 75, No. 2, 1978

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A mean value theorem for binary digits

Alan Stein and Ivan Ernest Stux

Vol. 75 (1978), No. 2, 565–577
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=1xα(k) xlog x∕2log 2. They then considered the iterates of α defined by αq = αq1 α and observed that Ar(x) = k=1xαr(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
Authors
Alan Stein
Ivan Ernest Stux