Order of zeros of Dedekind zeta functions
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- by Daniel Hu, Ikuya Kaneko, Spencer Martin and Carl Schildkraut PDF
- Proc. Amer. Math. Soc. 150 (2022), 5111-5120 Request permission
Abstract:
Answering a question of Browkin, we provide a new unconditional proof that the Dedekind zeta function of a number field $L$ has infinitely many nontrivial zeros of multiplicity at least 2 if $L$ has a subfield $K$ for which $L/K$ is a nonabelian Galois extension. We also extend this to zeros of order 3 when $Gal(L/K)$ has an irreducible representation of degree at least 3, as predicted by the Artin holomorphy conjecture.References
- S. A. Amitsur, Groups with representations of bounded degree. II, Illinois J. Math. 5 (1961), 198–205. MR 122893, DOI 10.1215/ijm/1255629818
- Hideo Aramata, Über die Teilbarkeit der Dedekindschen Zetafunktionen, Proc. Imp. Acad. Tokyo 9 (1933), no. 2, 31–34 (German). MR 1568340
- Jerzy Browkin, Multiple zeros of Dedekind zeta functions, Funct. Approx. Comment. Math. 49 (2013), no. 2, 383–390. MR 3161504, DOI 10.7169/facm/2013.49.2.15
- Richard Foote and V. Kumar Murty, Zeros and poles of Artin $L$-series, Math. Proc. Cambridge Philos. Soc. 105 (1989), no. 1, 5–11. MR 966135, DOI 10.1017/S0305004100001316
- Richard Foote and David Wales, Zeros of order $2$ of Dedekind zeta functions and Artin’s conjecture, J. Algebra 131 (1990), no. 1, 226–257. MR 1055006, DOI 10.1016/0021-8693(90)90173-L
- H. Heilbronn, Zeta-functions and $L$-functions, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., 1967, pp. 204–230. MR 0218327
- D. Hu, I. Kaneko, S. Martin, and C. Schildkraut, On the Mertens conjecture over number fields, 2021. e-prints, arXiv:2109.06665.
- Bertram Huppert, Monomiale Darstellung endlicher Gruppen, Nagoya Math. J. 6 (1953), 93–94 (German). MR 59273, DOI 10.1017/S0027763000017025
- Gareth A. Jones, Varieties and simple groups, J. Austral. Math. Soc. 17 (1974), 163–173. Collection of articles dedicated to the memory of Hanna Neumann, VI. MR 0344342, DOI 10.1017/S1446788700016748
- H. M. Stark, Some effective cases of the Brauer-Siegel theorem, Invent. Math. 23 (1974), 135–152. MR 342472, DOI 10.1007/BF01405166
- John G. Thompson, Nonsolvable finite groups all of whose local subgroups are solvable, Bull. Amer. Math. Soc. 74 (1968), 383–437. MR 230809, DOI 10.1090/S0002-9904-1968-11953-6
Additional Information
- Daniel Hu
- Affiliation: Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, New Jersey 08544-1000
- Email: danielhu@princeton.edu
- Ikuya Kaneko
- Affiliation: The Division of Physics, Mathematics and Astronomy, California Institute of Technology, 1200 E California Blvd, Pasadena, California 91125
- MR Author ID: 1356400
- ORCID: 0000-0003-4518-1805
- Email: ikuyak@icloud.com
- Spencer Martin
- Affiliation: UCLA Mathematics Department, Los Angeles, California 90095-1555
- ORCID: 0000-0003-4770-0007
- Email: stmartin@ucla.edu
- Carl Schildkraut
- Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139-4307
- Email: carlsc@mit.edu
- Received by editor(s): July 22, 2021
- Received by editor(s) in revised form: February 10, 2022
- Published electronically: June 17, 2022
- Additional Notes: The authors were supported by the National Science Foundation (Grants DMS 2002265 and DMS 205118), National Security Agency (Grant H98230-21-1-0059), the Thomas Jefferson Fund at the University of Virginia, and the Templeton World Charity Foundation.
- Communicated by: Amanda Folsom
- © Copyright 2022 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 150 (2022), 5111-5120
- MSC (2020): Primary 11R42; Secondary 20C15
- DOI: https://doi.org/10.1090/proc/16041