The Distribution of Sequences in Residue Classes

We prove that any set of integers A ⊂ [1, x] with |A| (log x)r lies in at least νA(p) p r r+1 many residue classes modulo most primes p (log x)r+1. (Here r is a positive constant.) This generalizes a result of Erdős and Ram Murty, who proved in connection with Artin’s conjecture on primitive roots that the integers below x which are multiplicatively generated by the coprime integers a1, . . . , ar (i.e. whose counting function is also c(log x)r) lie in at least p r r+1 +ε(p) residue classes, modulo most small primes p, where ε(p)→ 0, as p→∞. Let ordp(a) denote the order of a modulo p, where (a, p) = 1. A quantitative version of Artin’s conjecture on primitive roots states that for a fixed integer a, not a square, and not −1, there is a positive proportion of primes such that ordp(a) = p− 1. (See [7] for a survey.) In favour of this conjecture, Erdős proved in [2] that for all but o( y log y ) of the primes p ≤ y one has ordp(2) > p 1 2 . This improved upon the lower bound of ordp(2) > p for all δ < 12 , proved by Bundschuh; see [1]. It is also implicit in section 3 of Hooley’s work on Artin’s conjecture (see [5]) that for all but O( y (log y)3 ) primes p ≤ y one has ordp(a) √ p log p . Erdős announced at the end of his paper that the lower bound can be slightly sharpened to ordp(2) ≥ p 1 2 , where ε is any real function with limp→∞ ε(p) = 0. The details of this were provided by Erdős and Ram Murty in [3]. For related results see also Pappalardi [8]. For a more general situation, they implicitly consider the following. Let a1, . . ., ar be mutually coprime positive integers and let p be a prime with (p, a1a2 · · · ar) = 1. Let A1 be the semigroup of positive integers multiplicatively generated by the ai and let A = A1 ∩ [1, x]. Then the elements in A have the form a1 1 a β2 2 · · · ar r ≤ x, where the βi are nonnegative integers. Let f(p, a1, . . . , ar) denote the number of distinct residue classes modulo p which are needed to cover all elements of a ∈ A. Note that the number of powers of a below x is asymptotically ca log x, and that there are about ca1,... ,ar(log x) many integers below x which are generated Received by the editors March 9, 2001. 1991 Mathematics Subject Classification. Primary 11N69, 11N36; Secondary 11B50, 11A07.