Murmurations of Dirichlet characters
Oliver, T., Lee, K.-H. and Pozdnyakov, A. 2024. Murmurations of Dirichlet characters. International Mathematics Research Notices.
Oliver, T., Lee, K.-H. and Pozdnyakov, A. 2024. Murmurations of Dirichlet characters. International Mathematics Research Notices.
| Title | Murmurations of Dirichlet characters |
|---|---|
| Type | Journal article |
| Authors | Oliver, T., Lee, K.-H. and Pozdnyakov, A. |
| Abstract | We calculate murmuration densities for two families of Dirichlet characters. The first family contains complex Dirichlet characters normalized by their Gauss sums. Integrating the first density over a geometric interval yields a murmuration function compatible with experimental observations. The second family contains real Dirichlet characters weighted by a smooth function with compact support. We show that the second density exhibits a universality property analogous to Zubrilina's density for holomorphic newforms, and it interpolates the phase transition in the the 1-level density for a symplectic family of L-functions. |
| Journal | International Mathematics Research Notices |
| ISSN | 1687-0247 |
| 1073-7928 | |
| Year | 2024 |
| Publisher | Oxford University Press |
| Accepted author manuscript | File Access Level Open (open metadata and files) |
Murmurations of elliptic curves
Oliver, T., He, Y.-H., Lee, K.-H. and Pozdnyakov, A. 2024. Murmurations of elliptic curves. Experimental Mathematics. Advanced online publication. https://doi.org/10.1080/10586458.2024.2382361
Machine learning for number theory: unsupervised learning with L-functions
Oliver, T. 2024. Machine learning for number theory: unsupervised learning with L-functions. 8th International Congress on Mathematical Software (ICMS2024). Durham University 22 - 25 Jul 2024 Springer. https://doi.org/10.1007/978-3-031-64529-7_21
PCA, arithmetic, and murmurations
Oliver, T. 2024. PCA, arithmetic, and murmurations. Murmurations in Arithmetic. ICERM (Brown University) 06 - 08 Jul 2023 World Scientific Publishing. https://doi.org/10.1142/S2810939224400021
Preprint: Murmurations of Dirichlet Characters
Lee, K.-H., Oliver, T. and Pozdnyakov, A. 2023. Preprint: Murmurations of Dirichlet Characters. arXiv. https://doi.org/10.48550/arXiv.2307.00256
Machine Learning Class Numbers of Real Quadratic Fields
Oliver, T., Amir, M., He, Y.-H., Lee, K.-H. and Sultanow, E. 2023. Machine Learning Class Numbers of Real Quadratic Fields. International Journal of Data Science in the Mathematical Sciences. 1 (2), pp. 107-134. https://doi.org/10.1142/s2810939223500016
Counting points on elliptic curves
Oliver, T. and Wuthrich, C. 2023. Counting points on elliptic curves. LMS newsletter. 509, pp. 31-35. https://doi.org/10.1112/NLMS
Supervised learning of arithmetic invariants
Oliver, T. 2023. Supervised learning of arithmetic invariants. in: Yang-Hui He (ed.) Machine Learning in Pure Mathematics and Theoretical Physics World Scientific Publishing. pp. 331-363
Machine learning invariants of arithmetic curves
He, Y.-H., Lee, K.-H. and Oliver, T. 2023. Machine learning invariants of arithmetic curves. Journal of Symbolic Computation. 115, pp. 478-491. https://doi.org/10.1016/j.jsc.2022.08.017
Ratios of Artin L-functions
Hochfilzer, L. and Oliver, T. 2022. Ratios of Artin L-functions. Journal of Number Theory. 236, pp. 1-40. https://doi.org/10.1016/j.jnt.2021.07.007
Preprint: Murmurations of Elliptic Curves
He, Y.-H., Lee, K.-H., Oliver, T. and Pozdnyakov, A. 2022. Preprint: Murmurations of Elliptic Curves. arXiv. https://doi.org/10.48550/arxiv.2204.10140
Preprint: Machine Learning Class Numbers of Real Quadratic Fields
Amir, M., He, Y.-H., Lee, K.-H., Oliver, T. and Sultanow, E. 2022. Preprint: Machine Learning Class Numbers of Real Quadratic Fields. arXiv. https://doi.org/10.48550/arxiv.2209.09283
Preprint: Convergance of Kac-Moody Eisenstein Series over a Function Field
Lee, K.-H., Liu, D. and Oliver, T. 2022. Preprint: Convergance of Kac-Moody Eisenstein Series over a Function Field. arXiv. https://doi.org/10.48550/arxiv.2203.08628
Machine-learning the Sato-Tate conjecture
He, Y.-H., Lee, K.-H. and Oliver, T. 2022. Machine-learning the Sato-Tate conjecture. Journal of Symbolic Computation. 111, pp. 61-72. https://doi.org/10.1016/j.jsc.2021.11.002
Ratios of Artin L-functions
Hochfilzer, L. and Oliver, T. 2022. Ratios of Artin L-functions. Journal of Number Theory. 236, pp. 1-40. https://doi.org/10.1016/j.jnt.2021.07.007
Machine-learning number fields
He, Y.-H., Lee, K.-H. and Oliver, T. 2022. Machine-learning number fields. Mathematics, Computation and Geometry of Data. 2 (1), pp. 49-66. https://doi.org/10.4310/MCGD.2022.v2.n1.a2
Character expansion of Kac–Moody correction factors
Lee, K.-H., Liu, D and Oliver, T. 2021. Character expansion of Kac–Moody correction factors. Pacific Journal of Mathematics. 313 (1), pp. 159-183. https://doi.org/10.2140/pjm.2021.313.159
Preprint: Twisting moduli for gl(2)
Bedert, B., Cooper, G., Oliver, T. and Zhang, P. 2020. Preprint: Twisting moduli for gl(2). arXiv. https://doi.org/10.48550/arxiv.2003.02557
Preprint: Machine-Learning the Sato-Tate Conjecture
He, Y.-H., Lee, K.-H. and Oliver, T. 2020. Preprint: Machine-Learning the Sato-Tate Conjecture. arXiv. https://doi.org/10.48550/arxiv.2010.01213
Preprint: Machine-learning number fields
He, Y.-H., Lee, K.-H. and Oliver, T. 2020. Preprint: Machine-learning number fields. arXiv. https://doi.org/10.48550/arxiv.2011.08958
Preprint: Machine-learning arithmetic curves
He, Y.-H., Lee, K.-H. and Oliver, T. 2020. Preprint: Machine-learning arithmetic curves. arXiv. https://doi.org/10.48550/arxiv.2012.04084
Twisting moduli for GL(2)
Bedert, B., Cooper, G., Oliver, T. and Zhang, P. 2020. Twisting moduli for GL(2). Journal of Number Theory. 217, pp. 142-162. https://doi.org/10.1016/j.jnt.2020.04.008
Weil's converse theorem for Maass forms and cancellation of zeros
Oliver, T. and Neururer, M. 2020. Weil's converse theorem for Maass forms and cancellation of zeros. Acta Arithmetica. 196, pp. 387-422. https://doi.org/10.4064/aa190811-3-2
Preprint: Ratios of Artin L-functions
Hochfilzer, L. and Oliver, T. 2019. Preprint: Ratios of Artin L-functions. arXiv. https://doi.org/10.48550/arxiv.1910.02821
Preprint: Weil's Converse Theorem for Maass Forms and Cancellation of Zeros
Neururer, M. and Oliver, T. 2018. Preprint: Weil's Converse Theorem for Maass Forms and Cancellation of Zeros. arXiv. https://doi.org/10.48550/arxiv.1809.06586
A conjectural extension of Hecke’s converse theorem
Bettin, S., Bober, J., Booker, A., Conrey, B., Lee, M., Molteni, G., Oliver, T., Platt, D. and Steiner, R. 2018. A conjectural extension of Hecke’s converse theorem. The Ramanujan Journal. 47, pp. 659-684. https://doi.org/10.1007/s11139-017-9953-y
A conjectural extension of hecke's converse theorem
Bettin, S., Bober, J.W., Booker, A.R., Conrey, B., Lee, M., Molteni, G., Oliver, T., Platt, D.J. and Steiner, R.S. 2017. A conjectural extension of hecke's converse theorem. arXiv. https://doi.org/10.48550/arxiv.1704.02570
Notes on low degree L-data
Oliver, T. 2017. Notes on low degree L-data. Analytic Number Theory and Related Areas. Research Institute for Mathematical Sciences, Kyoto University 04 - 06 Nov 2015
Automorphicity and mean-periodicity
Oliver, T. 2017. Automorphicity and mean-periodicity. Journal of the Mathematical Society of Japan. 69 (1), pp. 25-51. https://doi.org/10.2969/jmsj/06910025
Zeta integrals on arithmetic surfaces
Oliver, T. 2016. Zeta integrals on arithmetic surfaces. St Petersburg Math. J.. 27, pp. 1003-1028. https://doi.org/10.1090/spmj/1432
Higher Dimensional Adeles, Mean-Periodicity and Zeta Functions of Arithmetic Surfaces
Oliver, T. 2014. Higher Dimensional Adeles, Mean-Periodicity and Zeta Functions of Arithmetic Surfaces. PhD thesis University of Nottingham School of Mathematical Sciences
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Accepted author manuscript
Under embargo until 02 Mar 2025