A generalization of the Hardy-Littlewood conjecture

Abstract

A famous conjecture of Hardy and Littlewood claims the subadditivity of the prime counting function, namely that π(x+y) ≤ π(x)+π(y) holds for all integers x, y ≥ 2, where π(x) is the number of primes not exceeding x. It is widely believed nowadays that this conjecture is not true since Hensley and Richards stunningly discovered an incompatibility with the prime k-tuples conjecture. Despite this drawback, here we generalize the subadditivity conjecture to subsets of prime numbers possessing a rich collection of preassigned structures. We show that subadditivity holds in this extended manner over certain ranges of the parameters which are wide enough to imply that it holds in an almost all sense. Under the prime k-tuples conjecture, very large values of convex combinations of the prime counting function are obtained infinitely often, thereby indicating a strong deviation of π(x) from being convex, even in a localized form. Finally, a Tauberian type condition is given for subsets of prime numbers which in turn implies an extension of a classical phenomenon, originally suggested by Legendre, about the asymptotically best fit functions to π(x) of the shape x/(log x − A). © 2022, Colgate University. All rights reserved.