Irregularity in sifted sequences

AbstractK. F. Roth (Acta Arith. 9 (1964), 257–260) considered the distribution of a sequence N of distinct positive integers not exceeding N among the residue classes for each modulus not exceeding Q. He showed that a certain variance was >ρ(1 − ρ) Q2N, where ρ was the density of the sequence, implying that N is not too evenly distributed among the residue classes in all subintervals of [1, N] unless ρ is almost 0 or 1. In this paper we consider a sifted sequence, one which is forbidden to enter certain residue classes, and enquire how evenly the sequence falls into the remaining residue classes for each modulus. Our main result shows that another variance lies between bounded multiples of ρ(1 − ρ) Q2NΛ, where NΛ is the Selberg upper bound for the number of members of N in [1, N] and ρNΛ is the actual number. The lower bound implies Roth's result in the unsifted case