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BIBTEX RIS

A mean value theorem for binary digits

1978; Mathematical Sciences Publishers; Volume: 75; Issue: 2 Linguagem: Inglês

10.2140/pjm.1978.75.565

ISSN

1945-5844

Autores

Alan H. Stein, Ivan Stux,

Tópico(s)

Advanced Topology and Set Theory

Resumo

This paper continues the investigation of the dyadically additive function a defined by a(n) = the number of l'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 2J.ia(fc)~ x Iogjc/21og2.They then considered the iterates of a defined by a q = a q -i°a and observed that A r (x) = 2I=i a r (k) is not asymptotic to any elementary function for r ^ 2.In this paper the function A 2 (x) will be examined more closely.Defining c(x) by A 2 (x) = c(x)x log log x/2 log 2, we will prove the following theorems.THEOREM 1.As x ranges over the positive integers, c(x) ranges densely over [1/2,3/2].Furthermore, given any c E [1/2,3/2], there is an explicit way to construct a sequence of integers x for which c(x)-*c as LEMMA 1.If x E.M, then 3

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