New quadratic polynomials with high densities of prime values

Abstract:

Hardy and Littlewood’s Conjecture F implies that the asymptotic density of prime values of the polynomials $f_A(x) = x^2 + x + A,$ $A \in \mathbb {Z}$, is related to the discriminant $\Delta = 1 - 4A$ of $f_A(x)$ via a quantity $C(\Delta ).$ The larger $C(\Delta )$ is, the higher the asymptotic density of prime values for any quadratic polynomial of discriminant $\Delta$. A technique of Bach allows one to estimate $C(\Delta )$ accurately for any $\Delta < 0$, given the class number of the imaginary quadratic order with discriminant $\Delta$, and for any $\Delta > 0$ given the class number and regulator of the real quadratic order with discriminant $\Delta$. The Manitoba Scalable Sieve Unit (MSSU) has shown us how to rapidly generate many discriminants $\Delta$ for which $C(\Delta )$ is potentially large, and new methods for evaluating class numbers and regulators of quadratic orders allow us to compute accurate estimates of $C(\Delta )$ efficiently, even for values of $\Delta$ 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, $C(\Delta )$ is larger than any previously known examples.