On the prime values represented by polynomials

Abstract

Many conjectures have been made concerning the infinitude of prime values assumed by the irreducible polynomials in Z[x]. The most general thus far is the Bateman-Horn conjecture. While the Bateman-Horn conjecture remains open, “on average” results have been given by Baier and Zhao [BZ2], [BZ4] for quadratic polynomials. The Hardy-Littlewood circle method is the primary tool used in [BZ2]. Here, we use the circle method to extend [BZ2] to cubic polynomials. To this end, we need to use the large sieve for cubic Dirichlet characters due to Baier and Young [BY], a large sieve for algebraic number fields due to M.N. Huxley [H], Artin reciprocity, and bounds for exponential sums. We also discuss what might be needed for higher degree polynomials. Furthermore, we discuss an application of the Bateman-Horn conjecture to the density of suitably normalized polynomial roots to prime moduli in the unit interval.