Omega results for cubic field counts via the Katz-Sarnak philosophy
Heilbronn Number Theory Seminar
16th February 2022, 4:00 pm – 5:00 pm
Online, Zoom
I will discuss recent joint work with P. Cho, Y. Lee and A. Södergren. Since the results of Davenport-Heilbronn, much work has been done to obtain a precise estimate for the number of cubic fields of discriminant at most X. This includes work of Belabas-Bhargava-Pomerance, Bhargava-Shankar-Tsimerman and Taniguchi-Thorne. In this talk I will present a negative result, which states that the GRH implies that the error term in this estimate cannot be too small. Our approach involves low-lying zeros of Dedekind zeta functions of cubic fields (first studied by Yang), and is strongly related to the Katz-Sarnak conjectures and the ratios conjecture of Conrey, Farmer and Zirnbauer. I will also present numerical support for our result.
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