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

  • Alina BucurEmail author
  • 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.


Riemann Zeta Function Riemann Hypothesis Automorphic Representation Arithmetic Formula Critical Strip 
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The authors of the paper would like to thank the organizers of the WINE conference. The conference was funded by CIRM, Microsoft research, Number theory foundation, NSF and Clay Mathematics Institute. Their support is gratefully acknowledged. The research of A.-M. E.-H. was supported by the Academy of Finland, grant no. 138337. The research of A.B. was supported by Simons Foundation Award #244988.


  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)]MathSciNetCrossRefzbMATHGoogle Scholar
  3. Bombieri, E., Lagarias, J.C.: Complements to Li’s criterion for the Riemann hypothesis. J. Number Theory 77, 274–287 (1999)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  10. Gelbart, S., Shahidi, F.: Boundedness of automorphic L-functions in vertical strips. J. Amer. Math. Soc. 14, 79–107 (2001)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  14. Jacquet, H., Shalika, J.A.: On Euler products and the classification of automorphic representations II. Am. J. Math. 103, 777–815 (1981b)MathSciNetCrossRefzbMATHGoogle Scholar
  15. Kaczorowski, J., Perelli, A.: On the structure of the Selberg class, I: 0 ≤ d ≤ 1. Acta Math. 182, 207–241 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  16. Lagarias, J.C.: Li’s coefficients for automorphic L-functions. Ann. Inst. Fourier 57, 1689–1740 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  17. Moeglin, C., Waldspurger, J.-L.: Le spectre résiduel de G L(n). Ann. Sci. École Norm. Sup. 22, 605–674 (1989)MathSciNetzbMATHGoogle Scholar
  18. Li, X.-J.: The positivity of a sequence of numbers and the Riemann hypothesis. J. Number Theory 65, 325–333 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  19. Li, X.-J.: Explicit formulas for Dirichlet and Hecke L-functions. Ill. J. Math. 48, 491–503 (2004)zbMATHGoogle Scholar
  20. Odžak, A., Smajlović, L.: On Li’s coefficients for the Rankin-Selberg L-functions. Ramanujan J. 21, 303–334 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  21. Odžak, A., Smajlović, L.: On asymptotic behavior of generalized Li coefficients in the Selberg class. J. Number Theory 131, 519–535 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  22. Rudnick, Z., Sarnak, P.: Zeros of principal L-functions and random matrix theory. Duke Math. J. 81, 269–322 (1996)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  25. Shahidi, F.: Fourier transforms of intertwining operators and Plancherel measures for G L(n). Am. J. Math. 106, 67–111 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  26. Shahidi, F.: Local coefficients as Artin factors for real groups. Duke Math. J. 52, 973–1007 (1985)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Alina Bucur
    • 1
    Email author
  • 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

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