On the Joint Distribution ofq-Additive Functions in Residue Classes

Abstract

LetSq(n) denote the sum of digits ofnin baseq. For given pairwise coprime basesq1,…,qℓand arbitrary residue classesaimodmi(i=1,mldr;,ℓ), we obtain an estimate with error termO(N1−δ) for the quantity#{0⩽n<N:Sqi(n)≡aimodmi(i=1,…,ℓ)},which extends results of J. Bésineau and establishes a conjecture of A. O. Gelfond. More generally, we obtain estimate of this type with eachSqireplaced by an arbitrary integer-valued completelyqi-additive function.