Ono invariants of imaginary quadratic fields with class number three

AbstractLet Ed(x) denote the “Euler polynomial” x2+x+(1−d)/4 if d≡1(mod4) and x2−d if d≡2,3(mod4). Set Ω(n) to be the number of prime factors (counting multiplicity) of the positive integer n. The Ono invariant Onod of K=Q(d) 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. In 2002 J. Cohen and J. Sonn conjectured that hd=3⇔Onod=3 and −d=p≡3(mod4) is a prime. They verified that the conjecture is true for p<1.5×107. Moreover, they proved that the conjecture holds for p>1017 assuming the extended Riemann Hypothesis. In this paper, we show that the conjecture holds for p⩽2.5×1013 by the aid of computer. And using a result of Bach, we also proved that the conjecture holds for p>2.5×1013 assuming the extended Riemann Hypothesis. In conclusion, we proved the conjecture is true assuming the extended Riemann Hypothesis