In this paper, we study the moments of central values of certain unitary families of Hecke -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 -functions.
Dans cet article, nous étudions les moments des valeurs centrales de certaines familles unitaires de fonctions 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 de Hecke.
Revised:
Accepted:
Published online:
Mots-clés : Hecke $L$-functions, Hecke characters
Peng Gao 1; Liangyi Zhao 2

@article{JTNB_2021__33_1_1_0, author = {Peng Gao and Liangyi Zhao}, title = {Moments and {One} level density of certain unitary families of {Hecke} $L$-functions}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {1--16}, publisher = {Soci\'et\'e Arithm\'etique de Bordeaux}, volume = {33}, number = {1}, year = {2021}, doi = {10.5802/jtnb.1149}, language = {en}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1149/} }
TY - JOUR AU - Peng Gao AU - Liangyi Zhao TI - Moments and One level density of certain unitary families of Hecke $L$-functions JO - Journal de théorie des nombres de Bordeaux PY - 2021 SP - 1 EP - 16 VL - 33 IS - 1 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1149/ DO - 10.5802/jtnb.1149 LA - en ID - JTNB_2021__33_1_1_0 ER -
%0 Journal Article %A Peng Gao %A Liangyi Zhao %T Moments and One level density of certain unitary families of Hecke $L$-functions %J Journal de théorie des nombres de Bordeaux %D 2021 %P 1-16 %V 33 %N 1 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1149/ %R 10.5802/jtnb.1149 %G en %F JTNB_2021__33_1_1_0
Peng Gao; Liangyi Zhao. Moments and One level density of certain unitary families of Hecke $L$-functions. Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 1, pp. 1-16. doi : 10.5802/jtnb.1149. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1149/
[1] Zeros of Dirichlet -functions, Ann. Sci. Éc. Norm. Supér., Volume 25 (1992) no. 5, pp. 567-615 | DOI | Numdam | MR | Zbl
[2] Gauss and Jacobi sums, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, 1998 | Zbl
[3] On moments of twisted -functions, Am. J. Math., Volume 139 (2017) no. 3, pp. 707-768 | DOI | MR | Zbl
[4] Non-vanishing of Dirichlet -functions at the central point, Int. J. Number Theory, Volume 8 (2012) no. 8, pp. 1855-1881 | DOI | MR | Zbl
[5] The Riemann hypothesis and Hilbert’s tenth problem, Mathematics and its Applications, 4, Gordon and Breach Science Publishers, 1965 | MR | Zbl
[6] The second moment of Dirichlet twists of Hecke -functions, Acta Arith., Volume 140 (2009) no. 1, pp. 57-65 | DOI | MR | Zbl
[7] One level density of low-lying zeros of families of -functions, Compos. Math., Volume 147 (2011) no. 1, pp. 1-18 | MR | Zbl
[8] Moments of Quadratic Hecke -functions of Imaginary Quadratic Number Fields, J. Number Theory, Volume 209 (2020), pp. 359-377 | MR | Zbl
[9] One level density of low-lying zeros of quadratic and quartic Hecke -functions, Can. J. Math., Volume 72 (2020) no. 2, pp. 427-454 | MR | Zbl
[10] The fourth power mean of Dirichlet’s -functions, Analysis, Volume 1 (1981) no. 1, pp. 25-32 | DOI | MR | Zbl
[11] Linear statistics of low-lying zeros of -functions, Q. J. Math, Volume 54 (2003) no. 3, pp. 309-333 | DOI | MR | Zbl
[12] Analytic Number Theory, Colloquium Publications, 53, American Mathematical Society, 2004 | MR | Zbl
[13] 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] Random matrices, Frobenius eigenvalues, and monodromy, Colloquium Publications, 45, American Mathematical Society, 1999 | MR | Zbl
[15] Zeroes of zeta functions and symmetries, Bull. Am. Math. Soc., Volume 36 (1999) no. 1, pp. 1-26 | DOI | Zbl
[16] Nonvanishing of Dirichlet -functions, Algebra Number Theory, Volume 10 (2016) no. 10, pp. 2081-2091 | DOI | MR | Zbl
[17] Multiplicative number theory. I. Classical theory, Cambridge Studies in Advanced Mathematics, 97, Cambridge University Press, 2007 | MR | Zbl
[18] On simple zeros of certain -series, Number theory (Banff, 1988), Walter de Gruyter, 1990, pp. 427-439 | MR | Zbl
[19] Algebraic number theory, Grundlehren der Mathematischen Wissenschaften, 322, Springer, 1999 | Zbl
[20] On the -analogues of some theorems in the theory of the Riemann -function, Proc. Lond. Math. Soc., Volume 32 (1931), pp. 273-311 | DOI | MR | Zbl
[21] 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] Nonvanishing of quadratic Dirichlet -functions at , Ann. Math., Volume 152 (2000) no. 2, pp. 447-488 | DOI | MR
[23] The fourth moment of Dirichlet -functions, Analytic Number Theory (Clay Mathematics Proceedings), Volume 7, American Mathematical Society, 2007, pp. 239-246 | MR | Zbl
[24] Non-vanishing of -functions attached to automorphic representations of over , J. Reine Angew. Math., Volume 474 (1996), pp. 1-24 | DOI | MR | Zbl
[25] The fourth moment of Dirichlet -functions, Ann. Math., Volume 173 (2011) no. 1, pp. 1-50 | DOI | MR | Zbl
Cited by Sources: