Multiplicity of nontrivial zeros of primitive 𝐿-functions via higher-level correlations

Gonçalves, Felipe; de Laat, David; Leijenhorst, Nando

doi:10.1090/mcom/4005

Multiplicity of nontrivial zeros of primitive $L$-functions via higher-level correlations
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by Felipe Gonçalves, David de Laat and Nando Leijenhorst;

Math. Comp. 94 (2025), 2041-2058

DOI: https://doi.org/10.1090/mcom/4005

Published electronically: August 14, 2024

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Abstract:

We give universal bounds on the fraction of nontrivial zeros having given multiplicity for $L$-functions attached to a cuspidal automorphic representation of $\mathrm {GL}_m/\mathbb {Q}$. For this, we apply the higher-level correlation asymptotic of Hejhal, Rudnick, and Sarnak in conjunction with semidefinite programming bounds.

References

Andrew R. Booker, Simple zeros of degree 2 $L$-functions, J. Eur. Math. Soc. (JEMS) 18 (2016), no. 4, 813–823. MR 3474457, DOI 10.4171/JEMS/603
Andrew R. Booker, Peter J. Cho, and Myoungil Kim, Simple zeros of automorphic $L$-functions, Compos. Math. 155 (2019), no. 6, 1224–1243. MR 3961571, DOI 10.1112/s0010437x19007279
Andrew R. Booker, Micah B. Milinovich, and Nathan Ng, Quantitative estimates for simple zeros of $L$-functions, Mathematika 65 (2019), no. 2, 375–399. MR 3902330, DOI 10.1112/s0025579318000530
Bence Borda, Lattice points in algebraic cross-polytopes and simplices, Discrete Comput. Geom. 60 (2018), no. 1, 145–169. MR 3807352, DOI 10.1007/s00454-017-9946-z
H. M. Bui and D. R. Heath-Brown, On simple zeros of the Riemann zeta-function, Bull. Lond. Math. Soc. 45 (2013), no. 5, 953–961. MR 3104987, DOI 10.1112/blms/bdt026
Emanuel Carneiro, Vorrapan Chandee, Friedrich Littmann, and Micah B. Milinovich, Hilbert spaces and the pair correlation of zeros of the Riemann zeta-function, J. Reine Angew. Math. 725 (2017), 143–182. MR 3630120, DOI 10.1515/crelle-2014-0078
Emanuel Carneiro, Andrés Chirre, and Micah B. Milinovich, Hilbert spaces and low-lying zeros of $L$-functions. part B, Adv. Math. 410 (2022), no. part B, Paper No. 108748, 48. MR 4498203, DOI 10.1016/j.aim.2022.108748
A. Y. Cheer and D. A. Goldston, Simple zeros of the Riemann zeta-function, Proc. Amer. Math. Soc. 118 (1993), no. 2, 365–372. MR 1132849, DOI 10.1090/S0002-9939-1993-1132849-0
Andrés Chirre, Felipe Gonçalves, and David de Laat, Pair correlation estimates for the zeros of the zeta function via semidefinite programming, Adv. Math. 361 (2020), 106926, 22. MR 4037496, DOI 10.1016/j.aim.2019.106926
H. Cohn, D. de Laat, and A. Salmon, Three-point bounds for sphere packing, Preprint, arXiv:2206.15373, 2022.
Henry Cohn and Noam Elkies, New upper bounds on sphere packings. I, Ann. of Math. (2) 157 (2003), no. 2, 689–714. MR 1973059, DOI 10.4007/annals.2003.157.689
J. B. Conrey, A. Ghosh, and S. M. Gonek, Mean values of the Riemann zeta-function with application to the distribution of zeros, Number theory, trace formulas and discrete groups (Oslo, 1987) Academic Press, Boston, MA, 1989, pp. 185–199. MR 993316
J. B. Conrey, A. Ghosh, and S. M. Gonek, Simple zeros of the Riemann zeta-function, Proc. London Math. Soc. (3) 76 (1998), no. 3, 497–522. MR 1616809, DOI 10.1112/S0024611598000306
J. B. Conrey and A. Ghosh, Simple zeros of the Ramanujan $\tau$-Dirichlet series, Invent. Math. 94 (1988), no. 2, 403–419. MR 958837, DOI 10.1007/BF01394330
D. de Laat and N. Leijenhorst, Solving clustered low-rank semidefinite programs arising from polynomial optimization, Preprint, arXiv:2202.12077, 2022.
A. V. Efimov, An analog of Rudin’s theorem for continuous radial positive definite functions of several variables, Proc. Steklov Inst. Math. 284 (2014), no. suppl. 1, S79–S86. MR 3452639, DOI 10.1134/S0081543814020072
A. de Faveri, Simple zeros of $GL(2)$ $L$-functions, Preprint, arXiv:2109.15311.
Karin Gatermann and Pablo A. Parrilo, Symmetry groups, semidefinite programs, and sums of squares, J. Pure Appl. Algebra 192 (2004), no. 1-3, 95–128. MR 2067190, DOI 10.1016/j.jpaa.2003.12.011
D. A. Goldston, S. M. Gonek, A. E. Özlük, and C. Snyder, On the pair correlation of zeros of the Riemann zeta-function, Proc. London Math. Soc. (3) 80 (2000), no. 1, 31–49. MR 1719184, DOI 10.1112/S0024611500012211
Rachel Greenfeld and Nir Lev, Fuglede’s spectral set conjecture for convex polytopes, Anal. PDE 10 (2017), no. 6, 1497–1538. MR 3678495, DOI 10.2140/apde.2017.10.1497
Victor Havin and Burglind Jöricke, The uncertainty principle in harmonic analysis, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 28, Springer-Verlag, Berlin, 1994. MR 1303780, DOI 10.1007/978-3-642-78377-7
Dennis A. Hejhal, On the triple correlation of zeros of the zeta function, Internat. Math. Res. Notices 7 (1994), 293ff., approx. 10 pp.}, issn=1073-7928, review= MR 1283025, doi=10.1155/S1073792894000334, DOI 10.1155/S1073792894000334
Henryk Iwaniec and Emmanuel Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. MR 2061214, DOI 10.1090/coll/053
Nir Lev and Máté Matolcsi, The Fuglede conjecture for convex domains is true in all dimensions, Acta Math. 228 (2022), no. 2, 385–420. MR 4448683, DOI 10.4310/acta.2022.v228.n2.a3
Micah B. Milinovich and Nathan Ng, Simple zeros of modular $L$-functions, Proc. Lond. Math. Soc. (3) 109 (2014), no. 6, 1465–1506. MR 3293156, DOI 10.1112/plms/pdu041
H. L. Montgomery, The pair correlation of zeros of the zeta function, Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972) Proc. Sympos. Pure Math., Vol. XXIV, Amer. Math. Soc., Providence, RI, 1973, pp. 181–193. MR 337821
Hugh L. Montgomery, Distribution of the zeros of the Riemann zeta function, Proceedings of the International Congress of Mathematicians (Vancouver, B.C., 1974) Canad. Math. Congr., Montreal, QC, 1975, pp. 379–381. MR 419378
Gábor Pataki, On the rank of extreme matrices in semidefinite programs and the multiplicity of optimal eigenvalues, Math. Oper. Res. 23 (1998), no. 2, 339–358. MR 1626662, DOI 10.1287/moor.23.2.339
Zeév Rudnick and Peter Sarnak, The $n$-level correlations of zeros of the zeta function, C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), no. 10, 1027–1032 (English, with English and French summaries). MR 1305671
Zeév Rudnick and Peter Sarnak, Zeros of principal $L$-functions and random matrix theory, Duke Math. J. 81 (1996), no. 2, 269–322. A celebration of John F. Nash, Jr. MR 1395406, DOI 10.1215/S0012-7094-96-08115-6
Jean-Pierre Serre, Linear representations of finite groups, Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, New York-Heidelberg, 1977. Translated from the second French edition by Leonard L. Scott. MR 450380, DOI 10.1007/978-1-4684-9458-7
Keiju Sono, A note on simple zeros of primitive Dirichlet $L$-functions, Bull. Aust. Math. Soc. 93 (2016), no. 1, 19–30. MR 3436011, DOI 10.1017/S0004972715000623
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, NJ, 1970. MR 290095
B. A. Venkov, On a class of Euclidean polyhedra, Vestnik Leningrad. Univ. Ser. Mat. Fiz. Him. 9 (1954), no. 2, 11–31 (Russian). MR 94790

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Mathematics of Computation

Multiplicity of nontrivial zeros of primitive $L$-functions via higher-level correlations
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Abstract: