The Sum-of-Digits Function of Squares

Abstract

The set of squares n2, n<2k, is considered and the sum of binary digits s(n2) is split up into two parts s[<k](n2)+s[⩾ k](n2), where s[<k](n2) = s(n2 mod 2k) collects the first k digits and s[⩾ k](n2) = s(n2/2k) collects the remaining digits. Very precise results on the distribution of s[<k](n2) and s[⩾ k](n2) are presented. For example, asymptotic formulae are provided for the numbers #{n < 2k: s[<k](n2) = m} and #{n< 2k : s[⩾ k](n2) = 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(n2) is asymptotically equidistributed in residue classes.