On τ-Li Coefficients for Rankin–Selberg L-Functions

  • Alina Bucur
  • Anne-Maria Ernvall-Hytönen
  • Almasa Odžak
  • Edva Roditty-Gershon
  • Lejla Smajlović
Part of the Association for Women in Mathematics Series book series (AWMS, volume 2)


The generalized τ-Li criterion for a certain zeta or L-function states that non-negativity of τ-Li coefficients associated to this function is equivalent to non-vanishing of this function in the region Re s > τ∕2. For τ ∈ [1, 2) and positive integers n, we define τ-Li coefficients \(\lambda _{n}(\pi \times \pi ',\tau )\) associated to Rankin–Selberg L-functions attached to convolutions of two cuspidal, unitary automorphic representations π and π′. We investigate their properties, including the archimedean and non-archimedean terms, and the asymptotic behavior of these terms.


  1. Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables. NBS Applied Mathematics Series 55, National Bureau of Standards, Washington, DC (1964)Google Scholar
  2. Bombieri, E., Ghosh, A.: Around Davenport-Heilbronn function. Uspekhi Math. Nauk 66, 15–66 (2011) [translated in Russ. Math. Surv. 66, 221–270 (2011)]MathSciNetCrossRefMATHGoogle Scholar
  3. Bombieri, E., Lagarias, J.C.: Complements to Li’s criterion for the Riemann hypothesis. J. Number Theory 77, 274–287 (1999)MathSciNetCrossRefMATHGoogle Scholar
  4. Bucur, A., Ernvall-Hytönen, A.-M., Odžak, A., Smajlović, L.: On a Li-type criteria for-zero free regions of certain Dirichlet series with real coefficients (in preparation)Google Scholar
  5. Cogdell, J.W.: L-functions and converse theorems for GL n, Automorphic forms and applications, IAS/Park City Math. Ser. 12, Amer. Math. Soc, Providence, RI, 2007, 97–177Google Scholar
  6. Davenport, D., Heilbronn, H.: On the zeros of certain Dirichlet series II. J. Lond. Math. Soc. 11, 307–312 (1936)MathSciNetCrossRefGoogle Scholar
  7. Droll, A.D.: Variations of Li’s criterion for an extension of the Selberg class. Ph.D. thesis, Queen’s University, Kingston (2012)Google Scholar
  8. Ernvall-Hytönen, A.-M., Odžak, A., Smajlović, L., Sušic, M.: On the modified Li criterion for a certain class of L-functions, J. Number Theory, doi:10.1016/j.jnt.2015.03.019 (in print)
  9. Freitas, P.: A Li-type criterion for zero-free half-planes of Riemann’s zeta function. J. Lond. Math. Soc. 73, 399–414 (2006)MathSciNetCrossRefMATHGoogle Scholar
  10. Gelbart, S., Shahidi, F.: Boundedness of automorphic L-functions in vertical strips. J. Amer. Math. Soc. 14, 79–107 (2001)MathSciNetCrossRefMATHGoogle Scholar
  11. Gelfand, I.M., Kazhdan, D.: Representation of the group G L(n, K), where K is a local field. In: I.M. Gelfand (ed.) Lie Groups and Their Representations, pp. 95–118. Wiley, New York (1974)Google Scholar
  12. Iwaniec, H., Kowalski, E.: Analytic Number Theory. American Mathematical Society Colloquium Publications, vol. 53. American Mathematical Society, Providence (2004)Google Scholar
  13. Jacquet, H., Shalika, J.A.: On Euler products and the classification of automorphic representations I. Am. J. Math. 103, 499–558 (1981a)MathSciNetCrossRefMATHGoogle Scholar
  14. Jacquet, H., Shalika, J.A.: On Euler products and the classification of automorphic representations II. Am. J. Math. 103, 777–815 (1981b)MathSciNetCrossRefMATHGoogle Scholar
  15. Kaczorowski, J., Perelli, A.: On the structure of the Selberg class, I: 0 ≤ d ≤ 1. Acta Math. 182, 207–241 (1999)MathSciNetCrossRefMATHGoogle Scholar
  16. Lagarias, J.C.: Li’s coefficients for automorphic L-functions. Ann. Inst. Fourier 57, 1689–1740 (2007)MathSciNetCrossRefMATHGoogle Scholar
  17. Moeglin, C., Waldspurger, J.-L.: Le spectre résiduel de G L(n). Ann. Sci. École Norm. Sup. 22, 605–674 (1989)MathSciNetMATHGoogle Scholar
  18. Li, X.-J.: The positivity of a sequence of numbers and the Riemann hypothesis. J. Number Theory 65, 325–333 (1997)MathSciNetCrossRefMATHGoogle Scholar
  19. Li, X.-J.: Explicit formulas for Dirichlet and Hecke L-functions. Ill. J. Math. 48, 491–503 (2004)MATHGoogle Scholar
  20. Odžak, A., Smajlović, L.: On Li’s coefficients for the Rankin-Selberg L-functions. Ramanujan J. 21, 303–334 (2010)MathSciNetCrossRefMATHGoogle Scholar
  21. Odžak, A., Smajlović, L.: On asymptotic behavior of generalized Li coefficients in the Selberg class. J. Number Theory 131, 519–535 (2011)MathSciNetCrossRefMATHGoogle Scholar
  22. Rudnick, Z., Sarnak, P.: Zeros of principal L-functions and random matrix theory. Duke Math. J. 81, 269–322 (1996)MathSciNetCrossRefMATHGoogle Scholar
  23. Sekatskii, S.K.: Generalized Bombieri-Lagarias’ theorem and generalized Li’s criterion (2013) [arXiv:1304.7895]Google Scholar
  24. Shahidi, F.: On certain L-functions. Am. J. Math. 103, 297–355 (1981)MathSciNetCrossRefMATHGoogle Scholar
  25. Shahidi, F.: Fourier transforms of intertwining operators and Plancherel measures for G L(n). Am. J. Math. 106, 67–111 (1984)MathSciNetCrossRefMATHGoogle Scholar
  26. Shahidi, F.: Local coefficients as Artin factors for real groups. Duke Math. J. 52, 973–1007 (1985)MathSciNetCrossRefMATHGoogle Scholar
  27. Shahidi, F.: A proof of Langlands’ conjecture on Plancherel measures. Complementary series for p-adic groups. Ann. Math. 132, 273–330 (1990)Google Scholar
  28. Smajlović, L.: On Li’s criterion for the Riemann hypothesis for the Selberg class. J. Number Theory 130, 828–851 (2010)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Alina Bucur
    • 1
  • Anne-Maria Ernvall-Hytönen
    • 2
  • Almasa Odžak
    • 3
  • Edva Roditty-Gershon
    • 4
  • Lejla Smajlović
    • 3
  1. 1.Department of MathematicsLa JollaUSA
  2. 2.Department of Mathematics and StatisticsUniversity of HelsinkiHelsinkiFinland
  3. 3.Department of MathematicsUniversity of SarajevoSarajevoBosnia & Herzegovina
  4. 4.Raymond and Beverly Sackler School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael

Personalised recommendations