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) ∼ xlogx∕2log2.
They then considered the iterates of α defined by αq= αq−1∘ α and observed
that Ar(x) =∑k=1xαr(k) is not asymptotic to any elementary function for
r ≧ 2.