The Difference between Consecutive Prime Numbers V

Abstract

Let p n denote the n th prime and let ε be any positive number. In 1938 (3) Ishowed that, for an infinity of values of n , where, for k ≧1, log k +1 x = log (log k x ) and log 1 x = log x . In a recent paper ( 4 ) Schönhage has shown that the constant ⅓ may be replaced by the larger number ½ e γ , where γ is Euler's constant; this is achieved by means of a more efficient selection of the prime moduli used. Schönhage uses an estimate of mine for the number B 1 of positive integers n ≦ u that consist entirely of prime factors p ≦ y , where Herer x is large and α and δ are positive constants to be chosen suitably.