New quadratic polynomials with high densities of prime values

Hardy and Littlewood’s Conjecture F implies that the asymptotic density of prime values of the polynomials fA(x) = x 2 + x + A, A ∈ Z, is related to the discriminant ∆ = 1 − 4A of fA(x) via a quantity C(∆). The larger C(∆) is, the higher the asymptotic density of prime values for any quadratic polynomial of discriminant ∆. A technique of Bach allows one to estimate C(∆) accurately for any ∆ < 0, given the class number of the imaginary quadratic order with discriminant ∆, and for any ∆ > 0 given the class number and regulator of the real quadratic order with discriminant ∆. The Manitoba Scalable Sieve Unit (MSSU) has shown us how to rapidly generate many discriminants ∆ for which C(∆) is potentially large, and new methods for evaluating class numbers and regulators of quadratic orders allow us to compute accurate estimates of C(∆) efficiently, even for values of ∆ with as many as 70 decimal digits. Using these methods, we were able to find a number of discriminants for which, under the assumption of the Extended Riemann Hypothesis, C(∆) is larger than any previously known examples.