The Sum-of-Digits Function of Squares

Abstract

Abstract. We consider the set of squares n 2 , n < 2 k , and split up the sum of binary digits s(n 2 ) into two parts s [<k] (n 2 ) + s [≥k] (n 2 ), where s [<k] (n 2 ) = s(n 2 mod 2 k ) collects the first k digits and s [≥k] (n 2 ) = s( n 2 /2 k ) 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 < 2 k : s [<k] (n 2 ) = m} and #{n < 2 k : s [≥k] (n 2 ) = 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(n 2 ) is asymptotically equidistributed in residue classes.

Sums of digits in q-ary expansions

“…By Lemma 3.1, we thus have (q − 1)|u and so (q − 1)|u 2 . By Lemma 3.1 again, we have (q − 1)|s q (u 2 ) and so (q − 1)|e 1 . Let e 3 = e 2 /(q − 1) and e 4 = e 1 /(q − 1) so that s q (u) = (q − 1)(k(m + 1) + n + e 3 ) and s q (u 2 ) = (q − 1)(n − mk + e 4 ).…”

“…Other work was done by Drmota and Rivat [1] in studying the binary digits of squares. They showed that for any k if one takes the set of squares n 2 such that n < 2 k , we have that the two sets of numbers #{n < 2 k : s ≥k (n 2 ) = m} and #{n < 2 k : s <k (n 2 ) = m} are asymptotically equidistributed in residue classes where s |<k| (n 2 ) = s 2 (n 2 mod 2 k ) represents the sum of the rightmost k digits in n 2 and s |≥k| (n 2 ) = s 2 (⌊n 2 /2 k ⌋) represents the sum of the remaining digits.…”

Sur un problème de Gelfond : la somme des chiffres des nombres premiers

“…La fonction f ainsi définie est clairement périodique de période q . Cette fonction tronquée apparaît dans un contexte différent dans [12] où Drmota et Rivat etudient certaines propriétés de f .n 2 / lorsque est de l'ordre de log n. LEMME 5. Pour tout " > 0, il existe une constante c."/ telle que pour tous , , entiers avec > 0, > 0 et 0 6 6 =2, pour tout r 2 ‫ޚ‬ avec jrj < q , le nombre E.r; ; ; / de couples d'entiers .m; n/ tels que q 1 < m 6 q , q 1 < n 6 q et…”

Block additive functions on the Gaussian integers

“…also [5] and [6]). We directly derive asymptotic expansions for moments (Corollary 2) and a refinement of the central limit theorem stated in Corollary 1, further a local limit theorem (Corollary 3), uniform distribution in residue classes (Corollary 4) and uniform distribution modulo 1 (Corollary 5).…”