Omega results for cubic field counts via lower-order terms in the one-level density
Journal article, 2022

In this paper, we obtain a precise formula for the one-level density of L-functions attached to non-Galois cubic Dedekind zeta functions. We find a secondary term which is unique to this context, in the sense that no lower-order term of this shape has appeared in previously studied families. The presence of this new term allows us to deduce an omega result for cubic field counting functions, under the assumption of the Generalised Riemann Hypothesis. We also investigate the associated L-functions Ratios Conjecture and find that it does not predict this new lower-order term. Taking into account the secondary term in Roberts’s conjecture, we refine the Ratios Conjecture to one which captures this new term. Finally, we show that any improvement in the exponent of the error term of the recent Bhargava–Taniguchi–Thorne cubic field counting estimate would imply that the best possible error term in the refined Ratios Conjecture is Oε(X−13+ε) . This is in opposition with all previously studied families in which the expected error in the Ratios Conjecture prediction for the one-level density is Oε(X−12+ε) .

Author

Peter J. Cho

Ulsan National Institute of Science and Technology (UNIST)

Daniel Fiorilli

University Paris-Saclay

Yoonbok Lee

Incheon National University

Anders Södergren

Chalmers, Mathematical Sciences, Algebra and geometry

Forum of Mathematics, Sigma

20505094 (eISSN)

Vol. 10 e80

Värdefördelning för L-funktioner och zetafunktioner

Swedish Research Council (VR) (2016-03759), 2017-01-01 -- 2020-12-31.

Subject Categories

Algebra and Logic

Mathematical Analysis

DOI

10.1017/fms.2022.70

More information

Latest update

11/1/2023