On the P Λ Problem

The well-known “H conjecture” (see [HaR]) states that n + 1 is prime infinitely many often. This is equivalent to the existence of infinitely many primes p satisfying {p1/2} < p. The current methods of analytic number theory are far from being sufficient to prove these conjectures. However, Kubilius [Kub] and Ankeny [Ank] proved already about fifty years ago that, assuming the truth of the Riemann Hypothesis for Hecke L-functions with Größencharacters over Q(i), p = n+m is infinitely many often prime withm ≪ log p. This implies that {