Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-27T22:05:43.206Z Has data issue: false hasContentIssue false
  • English
  • Français

On the zeros of Dirichlet $L$-functions

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

Published online by Cambridge University Press:  01 May 2018

Sami Omar ,
Raouf Ouni  and
Kamel Mazhouda
Sami Omar
Affiliation:
Department of Mathematics, King Khalid University, Abha 9004, Saudi Arabia Department of Mathematics, Faculty of Science of Tunis, 2092 Tunis, Tunisia email sami.omar@fst.rnu.tn
Raouf Ouni
Affiliation:
Department of Mathematics, Faculty of Science of Tunis, 2092 Tunis, Tunisia email raouf.ouni@fst.rnu.tn
Kamel Mazhouda
Affiliation:
Department of Mathematics, Faculty of Science of Monastir, 5000 Monastir, Tunisia email kamel.mazhouda@fsm.rnu.tn

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper [1], which was published online on 1 June 2011, has been retracted by agreement between the authors, the journal’s Editor-in-Chief Derek Holt, the London Mathematical Society and Cambridge University Press. The retraction was agreed to prevent other authors from using incorrect mathematical results. (In this paper, we compute and verify the positivity of the Li coefficients for the Dirichlet $L$-functions using an arithmetic formula established in Omar and Mazhouda, J. Number Theory 125 (2007) no. 1, 50–58; J. Number Theory 130 (2010) no. 4, 1109–1114. Furthermore, we formulate a criterion for the partial Riemann hypothesis and we provide some numerical evidence for it using new formulas for the Li coefficients.)

MSC classification

Primary: 11M06: $zeta (s)$ and $L(s, chi)$
Secondary: 11M26: Nonreal zeros of $zeta (s)$ and $L(s, chi)$; Riemann and other hypotheses 11M36: Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. Explicit formulas
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 14 , 2011 , pp. 140 - 154
Copyright
© The Author(s) 2011 

References

Omar, S., Ouni, R. and Mazhouda, K., ‘On the zeros of Dirichlet L-functions’, LMS J. Comput. Math. 14 (2011) 140–154; https://doi.org/10.1112/s1461157010000215.CrossRefGoogle Scholar
Brown, F., ‘Li’s criterion and zero-free regions of L-functions’, J. Number Theory 111 (2005) 1–32.CrossRefGoogle Scholar
Coffey, M., ‘Toward verification of the Riemann hypothesis: application of the Li criterion’, Math. Phys. Anal. Geom. 8 (2005) no. 3, 211–255.Google Scholar
Davenport, H., Multiplicative number theory (Springer, New York, 1980).Google Scholar
Gourdon, X., ‘The $10^{13}$ first zeros of the Riemann zeta function, and zeros computation at very large height’, available athttp://numbers.computation.free.fr/Constants/Miscellaneous/zetazeros1e13-1e24.pdf, October 2004.Google Scholar
Koepf, W. and Schmersau, D., ‘Bounded nonvanishing functions and Bateman functions’, Complex Var. 25 (1994) 237–259.Google Scholar
Li, X.-J., ‘The positivity of a sequence of numbers and the Riemann hypothesis’, J. Number Theory 65 (1997) no. 2, 325–333.CrossRefGoogle Scholar
Li, X.-J., ‘Explicit formulas for Dirichlet and Hecke L-functions’, Illinois J. Math. 48 (2004) no. 2, 491–503.Google Scholar
Maślanka, K., ‘Li’s criterion for the Riemann hypothesis — numerical approach’, Opuscula Math. 24 (2004) no. 1, 103–114.Google Scholar
Omar, S. and Bouanani, S., ‘Li’s criterion and the Riemann hypothesis for function fields’, Finite Fields Appl. 16 (2010) no. 6, 477–485.Google Scholar
Omar, S. and Mazhouda, K., ‘Le critère de Li et l’hypothèse de Riemann pour la classe de Selberg’, J. Number Theory 125 (2007) no. 1, 50–58.Google Scholar
Omar, S. and Mazhouda, K., ‘Corrigendum et addendum à “Le critère de Li et l’hypothèse de Riemann pour la classe de Selberg” [J. Number Theory 125 (2007) no. 1, 50–58]’, J. Number Theory 130 (2010) no. 4, 1109–1114.Google Scholar
Omar, S. and Mazhouda, K., ‘The Li criterion and the Riemann hypothesis for the Selberg class II’, J. Number Theory 130 (2010) no. 4, 1098–1108.Google Scholar
Omar, S., Ouni, R. and Mazhouda, K., ‘On the zeros of Hecke $L$ -functions’, Preprint, 2011.Google Scholar