Geodesic
On the One-Level Density Conjecture for Quadratic Dirichlet L-Functions
Canadian journal of mathematics, Tome 58 (2006) no. 4, pp. 843-858

Voir la notice de l'article provenant de la source Cambridge University Press

In a previous article, we studied the distribution of “low–lying” zeros of the family of quadratic Dirichlet $L$ –functions assuming the Generalized Riemann Hypothesis for all Dirichlet $L$ –functions. Even with this very strong assumption, we were limited to using weight functions whose Fourier transforms are supported in the interval (−2, 2). However, it is widely believed that this restriction may be removed, and this leads to what has become known as the One-Level Density Conjecture for the zeros of this family of quadratic $L$ -functions. In this note, we make use of Weil's explicit formula as modified by Besenfelder to prove an analogue of this conjecture.
DOI : 10.4153/CJM-2006-034-9
Mots-clés : 11M26 
Özlük, A. E.; Snyder, C. On the One-Level Density Conjecture for Quadratic Dirichlet L-Functions. Canadian journal of mathematics, Tome 58 (2006) no. 4, pp. 843-858. doi: 10.4153/CJM-2006-034-9
@article{10_4153_CJM_2006_034_9,
     author = {\"Ozl\"uk, A. E. and Snyder, C.},
     title = {On the {One-Level} {Density} {Conjecture} for {Quadratic} {Dirichlet} {L-Functions}},
     journal = {Canadian journal of mathematics},
     pages = {843--858},
     year = {2006},
     volume = {58},
     number = {4},
     doi = {10.4153/CJM-2006-034-9},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-2006-034-9/}
}
TY  - JOUR
AU  - Özlük, A. E.
AU  - Snyder, C.
TI  - On the One-Level Density Conjecture for Quadratic Dirichlet L-Functions
JO  - Canadian journal of mathematics
PY  - 2006
SP  - 843
EP  - 858
VL  - 58
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-2006-034-9/
DO  - 10.4153/CJM-2006-034-9
ID  - 10_4153_CJM_2006_034_9
ER  - 
%0 Journal Article
%A Özlük, A. E.
%A Snyder, C.
%T On the One-Level Density Conjecture for Quadratic Dirichlet L-Functions
%J Canadian journal of mathematics
%D 2006
%P 843-858
%V 58
%N 4
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-2006-034-9/
%R 10.4153/CJM-2006-034-9
%F 10_4153_CJM_2006_034_9

[1] [1] Barner, K., On A. Weil's explicit formula. J. Reine Angew. Math. 323(1981), 139–152. Google Scholar

[2] [2] Bellman, R., Analytic Number Theory, An Introduction. Mathematics Lecture Note Series 57, Benjamin Cummings, Reading, MA, 1980. Google Scholar

[3] [3] Besenfelder, H.-J., Die Weilsche “Explizite Formel” und temperierte Distributionen. J. Reine Angew. Math. 293/294(1977), 228–257. Google Scholar

[4] [4] Bentz, H.-J., Discrepancies in the distribution of prime numbers. J. Number Theory 15(1982), no. 2, 252–274. Google Scholar

[5] [5] Davenport, H., Multiplicative Number Theory. Second ed. Springer-Verlag, New York, 1980. Google Scholar

[6] [6] Iwaniec, H., Luo, W., and Sarnak, P. Low lying zeros of families of L-functions. Inst. Hautes Études Sci. Publ. Math. 91(2001), 55–131. Google Scholar

[7] [7] Katz, N. and Sarnak, P., Zeroes of zeta functions and symmetry. Bull. Amer. Math. Soc. 36(1999), no. 1, 1–26. Google Scholar

[8] [8] Katz, N. and Sarnak, P., Random Matrices, Frobenius Eigenvalues and Monodromy. American Mathematical Society Colloquium Publications 45, American Mathematical Society, Providence, RI, 1999. Google Scholar

[9] [9] Özlük, A., and Snyder, C., On the distribution of the nontrivial zeros of quadratic L-functions close to the real axis. Acta Arith. 91(1999), 209–228. Google Scholar

[10] [10] Soundararajan, K., Nonvanishing of quadratic Dirichlet L-functions at s = 1/2. Annals of Math. 152(2000), no. 2, 447–488. Google Scholar

Cité par Sources :