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Zeros of GL2 𝐿-functions on the critical line

  • Nickolas Andersen and Jesse Thorner EMAIL logo
Published/Copyright: February 2, 2021

Abstract

We use Levinson’s method and the work of Blomer and Harcos on the GL2 shifted convolution problem to prove that at least 6.96 % of the nontrivial zeros of the 𝐿-function of a GL2 automorphic form lie on the critical line.

MSC 2010: 11F66

Award Identifier / Grant number: DMS-1701638

Funding statement: This work began while the first author was funded by NSF grant DMS-1701638 and the second author was a postdoctoral researcher at Stanford University (funded by a NSF Mathematical Sciences Postdoctoral Fellowship).

Acknowledgements

All computations were performed using Mathematica 12. We thank the anonymous referee for the helpful comments.

  1. Communicated by: Valentin Blomer

References

[1] D. Bernard, Statistics on non-trivial zeros of modular L-functions, Thesis, University Blaise Pascal - Clermont-Ferrand II, 2013. Search in Google Scholar

[2] D. Bernard, Modular case of Levinson’s theorem, Acta Arith. 167 (2015), no. 3, 201–237. 10.4064/aa167-3-1Search in Google Scholar

[3] V. Blomer, Rankin–Selberg 𝐿-functions on the critical line, Manuscripta Math. 117 (2005), no. 2, 111–133. 10.1007/s00229-005-0557-2Search in Google Scholar

[4] V. Blomer and F. Brumley, On the Ramanujan conjecture over number fields, Ann. of Math. (2) 174 (2011), no. 1, 581–605. 10.4007/annals.2011.174.1.18Search in Google Scholar

[5] V. Blomer and G. Harcos, The spectral decomposition of shifted convolution sums, Duke Math. J. 144 (2008), no. 2, 321–339. 10.1215/00127094-2008-038Search in Google Scholar

[6] V. Blomer and G. Harcos, Twisted 𝐿-functions over number fields and Hilbert’s eleventh problem, Geom. Funct. Anal. 20 (2010), no. 1, 1–52. 10.1007/s00039-010-0063-xSearch in Google Scholar

[7] V. Blomer and D. Milićević, The second moment of twisted modular 𝐿-functions, Geom. Funct. Anal. 25 (2015), no. 2, 453–516. 10.1007/s00039-015-0318-7Search in Google Scholar

[8] B. Conrey, Zeros of derivatives of Riemann’s 𝜉-function on the critical line, J. Number Theory 16 (1983), no. 1, 49–74. 10.1016/0022-314X(83)90031-8Search in Google Scholar

[9] J. B. Conrey, More than two fifths of the zeros of the Riemann zeta function are on the critical line, J. Reine Angew. Math. 399 (1989), 1–26. 10.1515/crll.1989.399.1Search in Google Scholar

[10] J.-M. Deshouillers and H. Iwaniec, Kloosterman sums and Fourier coefficients of cusp forms, Invent. Math. 70 (1982/83), no. 2, 219–288. 10.1007/BF01390728Search in Google Scholar

[11] S. Gelbart and H. Jacquet, A relation between automorphic representations of GL(2) and GL(3), Ann. Sci. Éc. Norm. Supér. (4) 11 (1978), no. 4, 471–542. 10.24033/asens.1355Search in Google Scholar

[12] J. L. Hafner, Zeros on the critical line for Dirichlet series attached to certain cusp forms, Math. Ann. 264 (1983), no. 1, 21–37. 10.1007/BF01458048Search in Google Scholar

[13] J. L. Hafner, Zeros on the critical line for Maass wave form 𝐿-functions, J. Reine Angew. Math. 377 (1987), 127–158. 10.1515/crll.1987.377.127Search in Google Scholar

[14] J. Hoffstein and D. Ramakrishnan, Siegel zeros and cusp forms, Int. Math. Res. Not. IMRN 1995 (1995), no. 6, 279–308. 10.1155/S1073792895000225Search in Google Scholar

[15] H. Iwaniec, Spectral Methods of Automorphic Forms, 2nd ed., Grad Stud. Math. 53, American Mathematical Society, Providence, 2002. 10.1090/gsm/053Search in Google Scholar

[16] H. Iwaniec and E. Kowalski, Analytic Number Theory, Amer. Math. Soc. Colloq. Publ. 53, American Mathematical Society, Providence, 2004. 10.1090/coll/053Search in Google Scholar

[17] H. H. Kim, Functoriality for the exterior square of GL4 and the symmetric fourth of GL2, J. Amer. Math. Soc. 16 (2003), no. 1, 139–183. 10.1090/S0894-0347-02-00410-1Search in Google Scholar

[18] P. Kühn, N. Robles and D. Zeindler, On mean values of mollifiers and 𝐿-functions associated to primitive cusp forms, Math. Z. 291 (2019), no. 1–2, 661–709. 10.1007/s00209-018-2099-9Search in Google Scholar

[19] N. Levinson, More than one third of zeros of Riemann’s zeta-function are on σ=1/2, Adv. Math. 13 (1974), 383–436. 10.1016/0001-8708(74)90074-7Search in Google Scholar

[20] K. Pratt, N. Robles, A. Zaharescu and D. Zeindler, More than five-twelfths of the zeros of 𝜁 are on the critical line, Res. Math. Sci. 7 (2020), no. 2, Paper No. 2. 10.1007/s40687-019-0199-8Search in Google Scholar

[21] A. Selberg, On the zeros of Riemann’s zeta-function, Skr. Norske Vid.-Akad. Oslo I 1942 (1942), no. 10, 1–59. Search in Google Scholar

[22] M. P. Young, A short proof of Levinson’s theorem, Arch. Math. (Basel) 95 (2010), no. 6, 539–548. 10.1007/s00013-010-0199-9Search in Google Scholar

Received: 2020-04-29
Revised: 2020-12-21
Published Online: 2021-02-02
Published in Print: 2021-03-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

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