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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  [⩾ 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. Very precise results on the distribution of  s  [< k ] ( n  2 ) and  s  [⩾  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  [⩾  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.

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