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On the Li Coefficients for the Hecke L-functions

  • Sami Omar
  • Raouf Ouni
  • Kamel Mazhouda
Article

Abstract

In this paper, we compute and verify the positivity of the Li coefficients for the Hecke L-functions using an arithmetic formula established in Omar and Mazhouda, J. Number Theory 125(1), 50–58 (2007) and J. Number Theory 130(4), 1098–1108 (2010) and the Serre trace formula. Additional results are presented, including new formulas for the Li coefficients and a formulation of a criterion for the partial Riemann hypothesis. Basing on the numerical computations made below, we conjecture that these coefficients are increasing in n.

Keywords

Hecke L-functions Li’s criterion Riemann hypothesis 

Mathematics Subject Classifications (2010)

11M06 11M26 11M36 

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References

  1. 1.
    Bombieri, E., Lagarias, J.C.: Complements to Li’s criterion for the Riemann hypothesis. J. Number Theory 77(2), 274–287 (1999)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Brown, F.: Li’s criterion and zero-free regions of L-functions. J. Number Theory 111(1), 1–32 (2005)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Batut, C., Belabas, K., Bernardi, D., Cohen, H., Olivier, M.: Users Guide to PARI/GP, version 2.3.2, Bordeaux. http://pari.math.u-bordeaux.fr/ (2007)
  4. 4.
    Coffey, M.: Toward verification of the Riemann hypothesis: Application of the Li criterion. Math. Phys. Anal. Geom. 8(3), 211–255 (2005)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Coffey, M.: An explicit formula and estimations for Hecke L-functions: Applying the Li criterion. Int. J. Contemp. Math. Sci. 2(18), 859–870 (2007)MATHMathSciNetGoogle Scholar
  6. 6.
    Coffey, M.: On certain sums over the non-trivial zeta zeroes. Proc. R. Soc. A 466(2124), 3679–3692 (2010)CrossRefMATHMathSciNetADSGoogle Scholar
  7. 7.
    Davenport, H.: Multiplicative Number Theory. Springer, New York (1980)CrossRefMATHGoogle Scholar
  8. 8.
    Gourdon, X.: The 1013 first zeros of the Riemann zeta function, and zeros computation at very large height. available at http://numbers.computation.free.fr/Constants/Miscellaneous/zetazeros1e13-1e24.pdf, 2004 October
  9. 9.
    Koepf, W., Schmersau, D.: Bounded nonvanishing functions and Bateman functions. Complex Variables 25, 237–259 (1994)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Lagarias, J.C.: Li Coefficients for automorphic L-functions. Ann. Inst. Fourier (Grenoble) 57, 1689–1740 (2007)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Li, X.-J.: The positivity of a sequence of numbers and the Riemann hypothesis. J. Number Theory 65(2), 325–333 (1997)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Li, X.-J.: Explicit formulas for Dirichlet and Hecke L-functions. Illinois J. Math. 48(2), 491–503 (2004)MATHMathSciNetGoogle Scholar
  13. 13.
    Maślanka, K.: Li’s criterion for the Riemann hypothesis-numerical approach. Opuscula Math. 24(1), 103–114 (2004)MATHMathSciNetGoogle Scholar
  14. 14.
    Omar, S., Bouanani, S.: Li’s criterion and the Riemann hypothesis for function fields. Finite Fields Appl. 16(6), 477–485 (2010)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Omar, S., Mazhouda, K.: Le critr̄e de Li et l’hypothèse de Riemann pour la classe de Selberg. J. Number Theory 125(1), 50–58 (2007)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Omar, S., Mazhouda, K.: Corrigendum et addendum à, “Le critère de Li et l’hypothèse de Riemann pour la classe de Selberg” [J. Number Theory 125(1), 50–58 (2007)]. J. Number Theory 130(4), 1099–1114 (2010)Google Scholar
  17. 17.
    Omar, S., Mazhouda, K.: The Li criterion and the Riemann hypothesis for the Selberg class II. J. Number Theory 130(4), 1098–1108 (2010)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Omar, S., Ouni, R., Mazhouda, K.: On the zeros of Dirichlet L-functions. LMS J. Comput. Math. 14, 140–154 (2011)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Radziejewski, M.: A set of scripts for studying zeros of Hecke L-functions using the excellent Pari system. Radziejewski home page (www.staff.amu.edu.pl/maciejr/welcome.html)
  20. 20.
    Serre, J.-P.: Répartition asymptotique des valeurs propres de l’opérateur de Hecke T p. J. Am. Math. Soc. 10, 75–102 (1997)CrossRefMATHGoogle Scholar
  21. 21.
    Stein, W.A. et al.: Sage Mathematics Software (Version 4.3.5). The Sage Development Team, http://www.sagemath.org (2010)

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of MathematicsFaculty of Sciences of TunisTunisTunisia
  2. 2.Department of MathematicsFaculty of Sciences of MonastirMonastirTunisia

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