Rabinowitsch revisited

Artin prime producing quadratics

“…This is akin to HEATH-BROWN'S result that if there are many Siegel zeros, then the twin primes behave as expected. For more on the analytic aspects of prime-producing polynomials, see [8].…”

On primes represented by quadratic polynomials

“…holds for a positive proportion of integers A in the range R < A < 2R. They also proved in [19] that an asymptotic formula for P f (x), with f belonging to certain families of quadratic polynomials, holds for x in some range under the assumption of the existence of a Siegel zero for the relevant Dirichlet L-function. The methods used come from a paper of J.…”

“…G. Rabinowitsch [36] showed that n 2 + n + A is prime for 0 ≤ n ≤ A − 2 if and only if 4A − 1 is square-free and the ring of integers of the number field Q( √ 1 − 4A) has class number one. This question was further studied by A. Granville and R. A. Mollin in [19] and the works, particularly those of Mollin, referred to therein. It is most note-worthy that an upper bound for P f (x) of the order of magnitude predicted by (1.1) was proved in [19] unconditionally uniform in f , and uniform in x under the Riemann hypothesis for the Dirichlet L-function L(s, (D/·)).…”

“…This question was further studied by A. Granville and R. A. Mollin in [19] and the works, particularly those of Mollin, referred to therein. It is most note-worthy that an upper bound for P f (x) of the order of magnitude predicted by (1.1) was proved in [19] unconditionally uniform in f , and uniform in x under the Riemann hypothesis for the Dirichlet L-function L(s, (D/·)). Furthermore, it was shown unconditionally in [19] that for large R and N with R ε < N < √ R, # n ≤ N :…”

On Primes in Quadratic Progressions

“…It is most note-worthy that an upper bound of the order of magnitude predicted by (1.1) was proved by A. Granville and R. A. Mollin in [13] unconditionally uniform in the family of quadratic polynomials, and uniform in x under the Riemann hypothesis for a certain Dirichlet L-function. Furthermore, it is shown unconditionally in [13] that for large R and N with R ε < N < √ R, # n ≤ N : n 2 + n + A ∈ P ≍ L 1, 1 − 4A · −1 N log N holds for at least a postive proportion of integers A in the range R < A < 2R.…”