Rocky Mountain Journal of Mathematics

On the $\tau $-Li coefficients for automorphic $L$-functions

Kamel Mazhouda

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In this paper, we extend the Li coefficients for automorphic $L$-functions and the Li criterion for the Riemann hypothesis to yield a necessary and sufficient condition for the existence of zero-free strips for automorphic $L$-functions inside the critical strip. Next, we give an arithmetical and asymptotical formula for these coefficients. Finally, we show that there exists an entire function of exponential type that interpolates the extended Li coefficients (or the $\tau $-Li coefficients) at integer values. The results of this paper arise from ideas of the author~\cite {15}, Freitas~\cite {8}, Lagarias~\cite {10} and Odz$\breve {a}$k and Smajlovi$\grave {c}$~\cite {17}.

Article information

Source
Rocky Mountain J. Math., Volume 47, Number 6 (2017), 1987-2011.

Dates
First available in Project Euclid: 21 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1511254961

Digital Object Identifier
doi:10.1216/RMJ-2017-47-6-1987

Mathematical Reviews number (MathSciNet)
MR3725253

Zentralblatt MATH identifier
06816579

Subjects
Primary: 11M06: $\zeta (s)$ and $L(s, \chi)$ 11M36: Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. Explicit formulas 11M41: Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72}

Keywords
Automorphic $L$-functions Dirichlet series Li's criterion Riemann hypothesis

Citation

Mazhouda, Kamel. On the $\tau $-Li coefficients for automorphic $L$-functions. Rocky Mountain J. Math. 47 (2017), no. 6, 1987--2011. doi:10.1216/RMJ-2017-47-6-1987. https://projecteuclid.org/euclid.rmjm/1511254961


Export citation

References

  • K. Barner, On A. Weil's explicit formula, J. reine angew. Math. 323 (1981), 139–152.
    Zentralblatt MATH: 0446.12013
  • E. Bombieri and J.C. Lagarias, Complements to Li's criterion for the Riemann hypothesis, J. Num. Th. 77 (1999), 274–287.
    Mathematical Reviews (MathSciNet): MR1702145
    Zentralblatt MATH: 0972.11079
    Digital Object Identifier: doi:10.1006/jnth.1999.2392
  • F. Brown, Li's criterion and zero-free regions of $L$-functions, J. Num. Th. \bf111 (2005), 1–32.
    Zentralblatt MATH: 1154.11334
    Digital Object Identifier: doi:10.1016/j.jnt.2004.07.016
  • M. Coffey, An explicit formula and estimations for Hecke L-functions: Applying the Li criterion, Int. J. Contemp. Math. Sci. 2 (2007), 859–870.
    Zentralblatt MATH: 1214.11094
    Digital Object Identifier: doi:10.12988/ijcms.2007.07088
  • ––––, On certain sums over the non-trivial zeta zeroes, Proc. Roy Soc. Lond. \bf466 (2010), 3679–3692.
    Zentralblatt MATH: 1247.11108
    Digital Object Identifier: doi:10.1098/rspa.2010.0064
  • ––––, Toward verification of the Riemann hypothesis: Application of the Li criterion, Math. Phys. Anal. Geom. 8 (2005), 211–255.
    Zentralblatt MATH: 1097.11042
    Digital Object Identifier: doi:10.1007/s11040-005-7584-9
  • J.B. Conrey, The Riemann hypothesis, Not. Amer. Math. Soc. \bf50 (2003), 341–353.
    Zentralblatt MATH: 1160.11341
  • P. Freitas, A Li-type criterion for zero-free half-planes of Riemann's zeta function, J. Lond. Math. Soc. \bf73 (2006), 399–414.
    Mathematical Reviews (MathSciNet): MR2225494
    Zentralblatt MATH: 1102.11046
    Digital Object Identifier: doi:10.1112/S0024610706022599
  • H. Iwaniec and E. Kowalski, Analytic number theory, American Mathematical Society, Providence, RI, 2004.
    Zentralblatt MATH: 1059.11001
  • J. Lagarias, The Riemann hypothesis: Arithmetic and geometry, Clay Math. Proc. \bf6 (2006), 127–140.
    Zentralblatt MATH: 1119.11051
  • ––––, Li Coefficients for automorphic L-functions, Ann. Inst. Fourier (Grenoble) 57 (2007), 1689–1740.
  • X.-J. Li, The positivity of a sequence of numbers and the Riemann hypothesis, J. Num. Th. 65 (1997), 325–333.
    Zentralblatt MATH: 0884.11036
    Digital Object Identifier: doi:10.1006/jnth.1997.2137
  • ––––, Explicit formulas for Dirichlet and Hecke $L$-functions, Illinois J. Math. 48 (2004), 491–503.
  • K. Mazhouda, The saddle-point method and the generalized Li coefficients, Canad. Math. Bull. \bf54 (2011), 316–329.
    Zentralblatt MATH: 1294.11152
    Digital Object Identifier: doi:10.4153/CMB-2011-016-6
  • ––––, On the Li coefficients for the Dirichlet $L$-function, Monatsh. Math. 170 (2013), 405–423.
  • M.R. Murty, Problems in analytic number theory, Grad. Texts Math. vol No.?\qquad, Springer-Verlag, Berlin, 2001.
    Zentralblatt MATH: 0971.11001
  • A. Odzăk and L. Smajlovi\`c, On interpolation functions for generalized Li coefficients in the Selberg class, Int. J. Num. Th. \bf7 (2011), 771–792.
  • S. Omar and S. Bouanani, Li's criterion and the Riemann hypothesis for function fields, Finite Fields Appl. 16 (2010), 477–485.
    Zentralblatt MATH: 1222.11138
    Digital Object Identifier: doi:10.1016/j.ffa.2010.09.004
  • S. Omar and K. Mazhouda, Le critère de Li et l'hypothèse de Riemann pour la classe de Selberg, J. Num. Th. 125 (2007), 50–58.
  • ––––, Le critère de positivité de Li pour la classe de Selberg, Compt. Rend. Math. \bf345 (2007), 245–248.
  • ––––, Erratum et addendum à l'article, “Le critère de Li et l'hypothèse de Riemann pour la classe de Selberg," J. Num. Th. \bf130 (2010), 1109-1114.
  • ––––, The Li criterion and the Riemann hypothesis for the Selberg class, II, \bf130 (2010), 1098–1108.
    Zentralblatt MATH: 1188.11044
    Digital Object Identifier: doi:10.1016/j.jnt.2009.08.012
  • S. Omar, R. Ouni and K. Mazhouda, On the zeros of Dirichlet $L$-functions, LMS J. Comp. Math. 14 (2011), 140–154.
    Zentralblatt MATH: 1294.11144
    Digital Object Identifier: doi:10.1112/S1461157010000215
  • ––––, On the Li Coefficients for the Hecke L-functions, Math. Phys. Anal. Geom. 17 (2014), 67–81.
    Zentralblatt MATH: 1356.11060
    Digital Object Identifier: doi:10.1007/s11040-014-9141-x
  • Z. Rudnick and P. Sarnak, Zeros of principal L-functions and random matrix theory, Duke Math. J. \bf81 (1996), 269–322.
    Zentralblatt MATH: 0866.11050
    Digital Object Identifier: doi:10.1215/S0012-7094-96-08115-6
  • A. Selberg, Old and new conjectures and results about a class of Dirichlet series, in Collected papers 2, Springer-Verlag, New York, 1991.
    Zentralblatt MATH: 0787.11037
  • L. Smajlović, On Li's criterion for the Riemann hypothesis for the Selberg class, J. Num. Th. \bf130 (2010), 828–851.
    Zentralblatt MATH: 1188.11046
    Digital Object Identifier: doi:10.1016/j.jnt.2009.10.012

Rocky Mountain Mathematics Consortium

Rocky Mountain Journal of Mathematics

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more