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On The Ono Invariants of Imaginary Quadratic Fields

✍ Scribed by Joseph Cohen; Jack Sonn

Publisher
Elsevier Science
Year
2002
Tongue
English
Weight
127 KB
Volume
95
Category
Article
ISSN
0022-314X

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✦ Synopsis

Let E d ðxÞ denote the ''Euler polynomial'' x 2 þ x þ ð1 À dÞ=4 if d 1 ðmod 4Þ and x 2 À d if d 2; 3 ðmod 4Þ. Set OðnÞ ¼ the number of prime factors (counting multiplicity) of the positive integer n. The Ono invariant Ono d of K is defined to be maxfOðE d ðbÞÞ: b ¼ 0; 1; . . . jD d j=4 À 1g except when d ¼ À1; À3 in which case Ono d is defined to be 1. Finally, let

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 h d ¼ 2 , Ono d ¼ 2 for all negative integers d (Sasaki). Sasaki also proved that h d 5Ono d for all negative integers d. If h d ¼ 3, then necessarily Ono d ¼ 3 and Àd is a prime p 3 ðmod 4Þ. 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 Àd a prime p 3 ðmod 4Þ and Ono d ¼ 3. Moreover, using a result of Bach (which assumes the extended Riemann hypothesis), we find that the conjecture holds for all Àd ¼ prime p 3 ðmod 4Þ greater than 10 17 . Computer calculations so far show that the conjecture holds up to 1:5 Â 10 7 . 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 lim d!À1 Ono d ¼ 1.

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