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On the zeros of Dirichlet $L$ -functions

  • Sami Omar (a1) (a2), Raouf Ouni (a3) and Kamel Mazhouda (a4)
  • Please note a correction has been issued for this article.

Abstract

This paper [1], which was published online on 1 June 2011, has been retracted by agreement between the authors, the journal’s Editor-in-Chief Derek Holt, the London Mathematical Society and Cambridge University Press.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.)

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References

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1. Omar, S., Ouni, R. and Mazhouda, K., ‘On the zeros of Dirichlet L-functions’, LMS J. Comput. Math. 14 (2011) 140154; https://doi.org/10.1112/s1461157010000215.
2. Brown, F., ‘Li’s criterion and zero-free regions of L-functions’, J. Number Theory 111 (2005) 132.
3. Coffey, M., ‘Toward verification of the Riemann hypothesis: application of the Li criterion’, Math. Phys. Anal. Geom. 8 (2005) no. 3, 211255.
4. Davenport, H., Multiplicative number theory (Springer, New York, 1980).
5. 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.
6. Koepf, W. and Schmersau, D., ‘Bounded nonvanishing functions and Bateman functions’, Complex Var. 25 (1994) 237259.
7. Li, X.-J., ‘The positivity of a sequence of numbers and the Riemann hypothesis’, J. Number Theory 65 (1997) no. 2, 325333.
8. Li, X.-J., ‘Explicit formulas for Dirichlet and Hecke L-functions’, Illinois J. Math. 48 (2004) no. 2, 491503.
9. Maślanka, K., ‘Li’s criterion for the Riemann hypothesis — numerical approach’, Opuscula Math. 24 (2004) no. 1, 103114.
10. Omar, S. and Bouanani, S., ‘Li’s criterion and the Riemann hypothesis for function fields’, Finite Fields Appl. 16 (2010) no. 6, 477485.
11. 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, 5058.
12. 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, 11091114.
13. Omar, S. and Mazhouda, K., ‘The Li criterion and the Riemann hypothesis for the Selberg class II’, J. Number Theory 130 (2010) no. 4, 10981108.
14. Omar, S., Ouni, R. and Mazhouda, K., ‘On the zeros of Hecke $L$ -functions’, Preprint, 2011.
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LMS Journal of Computation and Mathematics
  • ISSN: -
  • EISSN: 1461-1570
  • URL: /core/journals/lms-journal-of-computation-and-mathematics
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A correction has been issued for this article: