On the Small Sieve . I . Sifting by Primes
نویسندگان
چکیده
The main object of the paper is to prove that if P is a set of primes with sum of reciprocals _cx, where c is a positive constant depending only on K. A lower estimate is given for c and a similar result is achieved in the case when the condition of primality is substituted by the weaker condition that any m elements of the sifting set are coprime. 1. INTRODUCTION For a set A of natural numbers let F(x,A) denote the number of natural with a positive constant c depending only on K. At first sight this may seem obvious ("easy to see," the first-named author wrote 13 ]), but it is not. The sieves of Brun and Selberg give this result only if the sifting primes all lie below xa, a < 1. The reason is that these sieves give a main term, which is the expected number of unsifted elements, and a remainder term. In our case the expectation is x Fl (1-1/p)-xe-K , where P runs over all sets of primes satisfying Our main aim is to prove that 1/p < K.
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