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On the One-Level Density Conjecture for Quadratic Dirichlet L-Functions

Published online by Cambridge University Press:  20 November 2018

A. E. Özlük  and
C. Snyder
A. E. Özlük
Affiliation:
Department of Mathematics and Statistics, University of Maine, Orono, Maine 04469, U.S.A. e-mail: [email protected]
C. Snyder
Affiliation:
Research Institute of Mathematics, Orono, ME 04473, U.S.A. e-mail: [email protected]
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Abstract

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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.

Keywords

11M26
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 58 , Issue 4 , 01 August 2006 , pp. 843 - 858
Copyright
Copyright © Canadian Mathematical Society 2006

References

[1] Barner, K., On A. Weil's explicit formula. J. Reine Angew. Math. 323(1981), 139–152.Google Scholar
[2] Bellman, R., Analytic Number Theory, An Introduction. Mathematics Lecture Note Series 57, Benjamin Cummings, Reading, MA, 1980.Google Scholar
[3] Besenfelder, H.-J., Die Weilsche “Explizite Formel” und temperierte Distributionen. J. Reine Angew. Math. 293/294(1977), 228–257.Google Scholar
[4] Bentz, H.-J., Discrepancies in the distribution of prime numbers. J. Number Theory 15(1982), no. 2, 252–274.Google Scholar
[5] Davenport, H., Multiplicative Number Theory. Second ed. Springer-Verlag, New York, 1980.Google Scholar
[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] Katz, N. and Sarnak, P., Zeroes of zeta functions and symmetry. Bull. Amer. Math. Soc. 36(1999), no. 1, 1–26.Google Scholar
[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] 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] Soundararajan, K., Nonvanishing of quadratic Dirichlet L-functions at s = 1/2. Annals of Math. 152(2000), no. 2, 447–488.Google Scholar