Unusually large gaps between consecutive primes

Abstract

ABSTRACT. Let G(x) The principal new tool used is a result of independent interest, namely, a mean value theorem for generalized twin primes lying in a residue class with a large modulus.

Landau’s problems on primes

“…Two improvements of the constant c 0 were achieved in the past 28 years. Maier and Pomerance (1990) showed this with c 0 = 1.31 . .…”

The ternary Goldbach problem with a prime and two isolated primes

“…For a long period of time the improvements of the result (1.2) only involved the constant C ( [12], [15], [18], [20]). A famous prize problem of Paul Erdős was the improvement of the order of magnitude of the function g. This was achieved only recently ( [6], [7], [13]).…”

THE DISTRIBUTION OF PRIMES: CONJECTURES vs. HITHERTO PROVABLES

“…Cramér's reasoning was based on the simple model that the probability of n being a prime is approximately 1/ log n, and that this can be taken into account independently for different integers (which is the basic flaw in Cramér's model; for example, n and n + 2 both being primes are not independent events: if n is even we automatically know that n + 2 can't be prime). Since Rankin's estimate (70), only the value of the constant c in (70) has been improved (see [80]). The estimates in (51) and (52), even under very strong conjectures for the zeta zeros, fall dismally short of (72).…”