Small gaps between primes or almost primes

Abstract

Abstract. Let p n denote the n th prime. Goldston, Pintz, and Yıldırım recently proved that lim infWe give an alternative proof of this result. We also prove some corresponding results for numbers with two prime factors. Let q n denote the n th number that is a product of exactly two distinct primes. We prove that lim infIf an appropriate generalization of the Elliott-Halberstam Conjecture is true, then the above bound can be improved to 6.

A smoothed GPY sieve

“…In all of the existing accounts [2]- [4] of the GPY sieve, it is assumed that (1.10) q≤x θ max (a,q)=1 max y≤x |E * (y; a, q)| ≪ x (log x) C 0 , with a certain absolute constant θ ∈ (0, 1) and an arbitrary fixed C 0 > 0; the implied constant depending only on C 0 .…”

“…Note that the case h / ∈ H in the last lemma, which is included in [2]- [4], is irrelevant for our present purpose. In fact, a combination of (1.10), (1.12), and (1.14) gives,…”

Small Gaps Between Almost Primes, the Parity Problem, and Some Conjectures of Erdos on Consecutive Integers

“…In the present work we shall show that the method which yielded the existence of short gaps between primes [11] and E 2 -numbers (numbers with exactly two distinct prime factors) with bounded differences [9], [10] is able to show a stronger variant of Conjectures C1-C3. In this variant, the parity problem is not bypassed but overcome.…”

THE DISTRIBUTION OF PRIMES: CONJECTURES vs. HITHERTO PROVABLES

“…The ideas involved in this work also yielded in [37] stronger variants of the Erdös-Mirsky conjecture: There are infinitely many integers n which simultaneously satisfy d(n) = d(n + 1), Ω(n) = Ω(n + 1), ω(n) = ω(n + 1), even by specifying the value of these functions along with generalizations to shifts n + b with an arbitrary positive integer b (here d(n), Ω(n), ω(n) denote respectively the number of positive integer divisors of n, the number of prime divisors of n counted with multiplicity, and the number of distinct prime divisors of n).…”