the sum-of-digits function of squares

Abstract

the set of squares $n^2$, $n<2^k$, is considered and the sum of binary digits $s(n^2)$ is split up into two parts $s_{[<k]}(n^2)+s_{[\ge k]}(n^2)$, where $s_{[<k]}(n^2) = s(n^2{\rm mod}2^k)$ collects the first $k$ digits and $s_{[\ge k]}(n^2) = s(\lfloor n^2/2^k\rfloor)$ collects the remaining digits. very precise results on the distribution of $s_{[<k]}(n^2)$ and $s_{[\ge k]}(n^2)$ are presented. for example, asymptotic formulae are provided for the numbers $\#\{n< 2^k{:} s_{[<k]}(n^2) = m\}$ and $\#\{n< 2^k {:} s_{[\ge k]}(n^2) = m\}$ and it is shown that these partial sum-of-digits functions are asymptotically equidistributed in residue classes. these results are prompted by a conjecture by gelfond saying that the (total) sum-of-digits function $s(n^2)$ is asymptotically equidistributed in residue classes.