Power and Exponential Sums of Digital Sums Related to Binomial Coefficient Parity

Let $F( x )$ be the number of odd numbers in the first x rows of Pascal’s triangle. Let $\theta = {( \log 3 ) / (\log 2 )} $. Let $\alpha = \lim \sup x^{-\theta} F( x )$ and $\beta = \lim \inf x^{-\theta} F( x )$. Then $0.72 \leqq \beta \leqq ( {\tfrac{9}{7}} )( {\tfrac{3} {4}} )^\theta \leqslant 0.815$ and $1 \leqq \alpha \leqq 1.052$. If $x = 2^{e_1 } + 2^{e_2 } + \cdots + 2^{e_r } $ where the $e_i $ are strictly decreasing, then $\sum\nolimits_{i = 1}^r {2^{i - 1} } 3^{e_i } $. These results are obtained from the known result that $F( x ) = \sum\nolimits_{n = 0}^{x - 1} {2^{B( n )} } $, where $B( n )$ is the number of ones in the binary expansion of n. The related sums $\sum\nolimits_{n \leqq x} {B^k ( n )} $ are shown to be of the form $x\{ {( \log x ) / ( {2\log 2} )} \}^k + O\{ x( \log x )^{k - 1} \}$; this is best possible. This curious history of digital sums and their estimates is briefly sketched.