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 θ = ( log 3 ) / (log 2 ). Let α = lim x-θ F( x ) and β = lim inf x-θ F( x ). Then 0.72 β ( 9/7 )( 3/4 )^θ 0.815 and 1 α 1.052. If x = 2e₁ + 2e₂ + ⋯ + 2eᵣ where the eᵢ are strictly decreasing, then ∑i = 1ʳ 2i - 1 3eᵢ. These results are obtained from the known result that F( x ) = ∑n = 0x - 1 2B( n ), where $B( n )$ is the number of ones in the binary expansion of n. The related sums ∑n x Bᵏ ( n ) are shown to be of the form x\ ( log x ) / ( 2log 2 ) \ᵏ + O\ x( log x )k - 1 \; this is best possible. This curious history of digital sums and their estimates is briefly sketched.