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A NOTE ON SIMPLE ZEROS OF PRIMITIVE DIRICHLET $L$-FUNCTIONS

Part of: Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  08 July 2015

KEIJU SONO
KEIJU SONO*
Affiliation:
Tokyo Denki University, Muzaigakuendai, Inzai, Chiba, Japan email souno@mail.dendai.ac.jp
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Abstract

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In this paper, by using the theory of reproducing kernel Hilbert spaces and the pair correlation formula constructed by Chandee et al. [‘Simple zeros of primitive Dirichlet $L$-functions and the asymptotic large sieve’, Q. J. Math.65(1) (2014), 63–87], we prove that at least 93.22% of low-lying zeros of primitive Dirichlet $L$-functions are simple in a proper sense, under the assumption of the generalised Riemann hypothesis.

Keywords

primitive Dirichlet L-functionssimple zerospair correlation formulareproducing kernel

MSC classification

Primary: 11M26: Nonreal zeros of $zeta (s)$ and $L(s, chi)$; Riemann and other hypotheses
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 93 , Issue 1 , February 2016 , pp. 19 - 30
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

References

Carneiro, E., Chandee, V., Littmann, F. and Milinovich, M. B., ‘Hilbert spaces and the pair correlation of zeros of the Riemann zeta-function’, J. reine angew. Math., to appear.Google Scholar
Chandee, V., Lee, Y., Liu, S.-C. and Radziwiłł, M., ‘Simple zeros of primitive Dirichlet L-functions and the asymptotic large sieve’, Q. J. Math. 65(1) (2014), 63–87.CrossRefGoogle Scholar
Cheer, A. Y. and Goldston, D. A., ‘Simple zeros of the Riemann zeta-function’, Proc. Amer. Math. Soc. 118(2) (1993), 365–372.CrossRefGoogle Scholar
Conrey, B., Iwaniec, H. and Soundararajan, K., ‘Asymptotic large sieve’, Preprint, arXiv:1105.1176, 2011.Google Scholar
Gallagher, P. X., ‘Pair correlation of zeros of the zeta function’, J. reine angew. Math. 362 (1985), 72–86.Google Scholar
Montgomery, H. L., ‘The pair correlation of zeros of the zeta function’, Proc. Sympos. Pure Math 24 (1973), 181–193.CrossRefGoogle Scholar
Montgomery, H. L., ‘Distribution of the zeros of the Riemann zeta function’, Proc. Int. Congr. Math., Vancouver, 21–29 August 1974 (ed. R. D. James) (International Congress of Mathematicians, Vancouver, 1974), 379–381.Google Scholar
Özlük, A. E., ‘On the q-analogue of the pair correlation conjecture’, J. Number Theory 59 (1996), 319–351.CrossRefGoogle Scholar