The Sum of Digits Function of Squares

We consider the set of squares n2, n < 2k, and split up the sum of binary digits s(n2) into two parts s[<k](n 2) + s[≥k](n 2), where s[<k](n 2) = s(n2 mod 2k) collects the first k digits and s[≥k](n 2) = s(bn2/2kc) collects the remaining digits. We present very precise results on the distribution on s[<k](n 2) and s[≥k](n 2). For example, we provide asymptotic formulas for the numbers #{n < 2k : s[<k](n) = m} and #{n < 2k : s[≥k](n) = m} and show that these partial sum of digits functions are asymptotically equidistributed in residue classes. These results are motivated by a conjecture by Gelfond [11] saying that the (total) sum of digits function s(n2) is asymptotically equidistributed in residue classes.