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Cho, Perter J.
Lab for Number Theory and L-functions
Research Interests
  • Number theory, artin L-functions, extreme values, random matrix, zeros of L-functions

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

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Title
Omega results for cubic field counts via lower-order terms in the one-level density
Author
Cho, Perter J.Fiorilli, DanielLee, YoonbokSodergren, Anders
Issue Date
2022-09
Publisher
Cambridge University Press
Citation
FORUM OF MATHEMATICS SIGMA, v.10, pp.e80
Abstract
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 s 𝑂𝜀 (𝑋− 1/3 +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− 1/2 +𝜀).
URI
https://scholarworks.unist.ac.kr/handle/201301/59084
DOI
10.1017/fms.2022.70
ISSN
2050-5094
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