Moments and One level density of certain unitary families of Hecke L-functions
Peng Gao1 ; Liangyi Zhao2
1 School of Mathematical Sciences Beihang University Beijing 100191, China
2 School of Mathematics and Statistics University of New South Wales Sydney NSW 2052, Australia
Journal de théorie des nombres de Bordeaux, Tome 33 (2021) no. 1, pp. 1-16.

Dans cet article, nous étudions les moments des valeurs centrales de certaines familles unitaires de fonctions L de Hecke sur le corps des rationnels de Gauss et prouvons un résultat quantitatif de non-annulation de leurs valeurs centrales. Nous établissons aussi un résultat de densité portant sur les petits zéros dans ces familles de fonctions L de Hecke.

In this paper, we study the moments of central values of certain unitary families of Hecke L-functions of the Gaussian field, and establish quantitative non-vanishing result for their central values. We also establish a one level density result for the low-lying zeros of these families of Hecke L-functions.

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DOI : 10.5802/jtnb.1149
Classification : 11M41, 11L40
Mots clés : Hecke $L$-functions, Hecke characters
Peng Gao 1 ; Liangyi Zhao 2

1 School of Mathematical Sciences Beihang University Beijing 100191, China
2 School of Mathematics and Statistics University of New South Wales Sydney NSW 2052, Australia
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Peng Gao; Liangyi Zhao. Moments and One level density of certain unitary families of Hecke $L$-functions. Journal de théorie des nombres de Bordeaux, Tome 33 (2021) no. 1, pp. 1-16. doi : 10.5802/jtnb.1149. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1149/

[1] Ramachandran Balasubramanian; V. Kumar Murty Zeros of Dirichlet L-functions, Ann. Sci. Éc. Norm. Supér., Volume 25 (1992) no. 5, pp. 567-615 | DOI | Numdam | MR | Zbl

[2] Bruce C. Berndt; Ronald J. Evans; Kenneth S. Williams Gauss and Jacobi sums, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, 1998 | Zbl

[3] Valentin Blomer; Étienne Fouvry; Emmanuel Kowalski; Philippe Michel; Djordje Milićević On moments of twisted L-functions, Am. J. Math., Volume 139 (2017) no. 3, pp. 707-768 | DOI | MR | Zbl

[4] Hung M. Bui Non-vanishing of Dirichlet L-functions at the central point, Int. J. Number Theory, Volume 8 (2012) no. 8, pp. 1855-1881 | DOI | MR | Zbl

[5] Sarvadaman Chowla The Riemann hypothesis and Hilbert’s tenth problem, Mathematics and its Applications, 4, Gordon and Breach Science Publishers, 1965 | MR | Zbl

[6] Peng Gao; Rizwanur Khan; Guillaume Ricotta The second moment of Dirichlet twists of Hecke L-functions, Acta Arith., Volume 140 (2009) no. 1, pp. 57-65 | DOI | MR | Zbl

[7] Peng Gao; Liangyi Zhao One level density of low-lying zeros of families of L-functions, Compos. Math., Volume 147 (2011) no. 1, pp. 1-18 | MR | Zbl

[8] Peng Gao; Liangyi Zhao Moments of Quadratic Hecke L-functions of Imaginary Quadratic Number Fields, J. Number Theory, Volume 209 (2020), pp. 359-377 | MR | Zbl

[9] Peng Gao; Liangyi Zhao One level density of low-lying zeros of quadratic and quartic Hecke L-functions, Can. J. Math., Volume 72 (2020) no. 2, pp. 427-454 | MR | Zbl

[10] David R. Heath-Brown The fourth power mean of Dirichlet’s L-functions, Analysis, Volume 1 (1981) no. 1, pp. 25-32 | DOI | MR | Zbl

[11] C. P. Hughes; Zeév Rudnick Linear statistics of low-lying zeros of L-functions, Q. J. Math, Volume 54 (2003) no. 3, pp. 309-333 | DOI | MR | Zbl

[12] Henryk Iwaniec; Emmanuel Kowalski Analytic Number Theory, Colloquium Publications, 53, American Mathematical Society, 2004 | MR | Zbl

[13] Henryk Iwaniec; Peter Sarnak Dirichlet L-functions at the central point, Number theory in progress. Volume 2: Elementary and analytic number theory, Walter de Gruyter, 1999, pp. 941-952 | Zbl

[14] Nicholas Katz; Peter Sarnak Random matrices, Frobenius eigenvalues, and monodromy, Colloquium Publications, 45, American Mathematical Society, 1999 | MR | Zbl

[15] Nicholas Katz; Peter Sarnak Zeroes of zeta functions and symmetries, Bull. Am. Math. Soc., Volume 36 (1999) no. 1, pp. 1-26 | DOI | Zbl

[16] Rizwanur Khan; Hieu T. Ngo Nonvanishing of Dirichlet L-functions, Algebra Number Theory, Volume 10 (2016) no. 10, pp. 2081-2091 | DOI | MR | Zbl

[17] Hugh L. Montgomery; Robert C. Vaughan Multiplicative number theory. I. Classical theory, Cambridge Studies in Advanced Mathematics, 97, Cambridge University Press, 2007 | MR | Zbl

[18] M. Ram Murty On simple zeros of certain L-series, Number theory (Banff, 1988), Walter de Gruyter, 1990, pp. 427-439 | MR | Zbl

[19] Jürgen Neukirch Algebraic number theory, Grundlehren der Mathematischen Wissenschaften, 322, Springer, 1999 | Zbl

[20] R. E. A. C. Paley On the k-analogues of some theorems in the theory of the Riemann ζ-function, Proc. Lond. Math. Soc., Volume 32 (1931), pp. 273-311 | DOI | MR | Zbl

[21] Werner Schaal Obere und untere Abschätzungen in algebraischen Zahlkörpern mit Hilfe des linearen Selbergschen Siebes, Acta Arith., Volume 13 (1968), pp. 267-313 | DOI | MR | Zbl

[22] Kannan Soundararajan Nonvanishing of quadratic Dirichlet L-functions at s=1 2, Ann. Math., Volume 152 (2000) no. 2, pp. 447-488 | DOI | MR

[23] Kannan Soundararajan The fourth moment of Dirichlet L-functions, Analytic Number Theory (Clay Mathematics Proceedings), Volume 7, American Mathematical Society, 2007, pp. 239-246 | MR | Zbl

[24] Tomasz Stefanicki Non-vanishing of L-functions attached to automorphic representations of GL (2) over Q, J. Reine Angew. Math., Volume 474 (1996), pp. 1-24 | DOI | MR | Zbl

[25] Matthew P. Young The fourth moment of Dirichlet L-functions, Ann. Math., Volume 173 (2011) no. 1, pp. 1-50 | DOI | MR | Zbl

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