On The Ono Invariants of Imaginary Quadratic Fields

Abstract

Let Ed(x) denote the “Euler polynomial” x2+x+(1−d)/4 if d≡1 (mod 4) and x2−d if d≡2,3 (mod 4). Set Ω(n)=the number of prime factors (counting multiplicity) of the positive integer n. The Ono invariantOnod of K is defined to be max{Ω(Ed(b)): b=0,1,…∣Δd∣/4−1∣ except when d=−1,−3 in which case Onod is defined to be 1. Finally, let hd=hK denote the class number of K=Q(d). It is known that hd=1⇔Onod=1 for all negative integers d (Frobenius–Rabinowitch). We improve a result of Chowla–Cowles–Cowles and characterize imaginary quadratic fields of class number one in terms of least prime quadratic residues. This yields an elementary characterization of imaginary quadratic fields with Ono invariant equal to one. It is also known that hd=2⇔Onod=2 for all negative integers d (Sasaki). Sasaki also proved that hd⩾Onod for all negative integers d. If hd=3, then necessarily Onod=3 and −d is a prime p≡3 (mod 4). Computer calculations support the conjecture that the converse holds. We prove in this paper that this conjecture fails for at most finitely many d. We will show in fact that there are only finitely many d with −d a prime p≡3 (mod 4) and Onod=3. Moreover, using a result of Bach (which assumes the extended Riemann hypothesis), we find that the conjecture holds for all −d=prime p≡3 (mod 4) greater than 1017. Computer calculations so far show that the conjecture holds up to 1.5×107. Another conditional result, derived from Lagarias–Odlyzko (Effective versions of the Chebotarev density theorem, in: “Algebraic Number Fields” (A. Frohlich, Ed.), Academic Press, London, 1977), which assumes GRH, is limd→−∞Onod=∞.