New equidistribution estimates of Zhang type

Abstract

Abstract. We prove distribution estimates for primes in arithmetic progressions to large smooth squarefree moduli, with respect to congruence classes obeying Chinese Remainder Theorem conditions, obtaining an exponent of distribution 1 2`7 300 .

Primitive Root Bias for Twin Primes

“…The first Hardy-Littlewood conjecture asserts that π 2 (x) ∼ 2C 2 is the twin primes constant [5]. A simpler expression that is asymptotically equivalent to (1.1) is 2C 2 x/(log x) 2 . A casual inspection (see Table 1) suggests that if p and p+2 are primes and p 5, then p has at least as many primitive roots as p + 2; that is, ϕ(p − 1) ϕ(p + 1).…”

The Bateman–Horn conjecture: Heuristic, history, and applications

“…Here are a few examples of Cunningham chains of the first kind (2,5,11,23,47), (3,7), (89, 179, 359, 719, 1439, 2879), and of the second kind (2, 3, 5), (7,13), (19,37,73).…”

Almost-prime values of polynomials at prime arguments

“…Recent work of Zhang [8] and Polymath [1] has given an improved level of distribution for the primes in arithmetic progressions to smooth moduli. This could be used to slightly improve our lower bound for S(A, z) by means of the Buchstab identity If w is chosen to be a suitably small power of x, then the results of Zhang and Polymath would apply to the remainder term when estimating S(A, w), thereby enabling us to sieve beyond √…”