Glasnik Matematicki, Vol. 51, No. 1 (2016), 23-44.

EULER-STIELTJES CONSTANTS FOR THE RANKIN-SELBERG L-FUNCTION AND WEIGHTED SELBERG ORTHOGONALITY

Almasa Odžak and Lejla Smajlović

Department of Mathematics, University of Sarajevo, Zmaja od Bosne 35, 71 000 Sarajevo, Bosnia and Herzegovina
e-mail: almasa@pmf.unsa.ba
e-mail: lejlas@pmf.unsa.ba

Abstract.   Let E be Galois extension of Q of finite degree and let π and π' be two irreducible automorphic unitary cuspidal representations of GLm(EA) and GLm'(EA), respectively. We prove an asymptotic formula for computation of coefficients γπ,π'(k) in the Laurent (Taylor) series expansion around s=1 of the logarithmic derivative of the Rankin-Selberg L-function L(s, π × π') under assumption that at least one of representations π, π' is self-contragredient and show that coefficients γπ,π'(k) are related to weighted Selberg orthogonality. We also replace the assumption that at least one of representations π and π' is self-contragredient by a weaker one.

2010 Mathematics Subject Classification.   11M26, 11S40.

Key words and phrases.   Euler-Stieltjes constants, Rankin-Selberg L-function, weighted Selberg orthogonality.


Full text (PDF) (access from subscribing institutions only)

DOI: 10.3336/gm.51.1.03


References:

  1. M. Avdispahić and L. Smajlović, Euler constants for a Fuchsian group of the first kind, Acta Arith. 131 (2008), 125-143.
    MathSciNet     CrossRef

  2. M. Avdispahić and L. Smajlović, On the Selberg orthogonality for automorphic L-functions, Arch. Math. 94 (2010), 147-154.
    MathSciNet     CrossRef

  3. W. E. Briggs, Some constants associated with the Riemann zeta-function, Mich. Math. J. 3 (1955-56), 117-121.
    MathSciNet     CrossRef

  4. M. W. Coffey, New results on the Stieltjes constants: Asymptotic and exact evaluation, J. Math. Anal. Appl. 317 (2006), 603-612.
    MathSciNet     CrossRef

  5. M. W. Coffey, Series representations for the Stieltjes constants, Rocky Mountain J. Math. 44 (2014), 443-477.
    MathSciNet     CrossRef

  6. J. W. Cogdell, L functions and converse theorems for GLn, Automorphic forms and applications, 97-177, IAS/Park City Math. Ser., 12, Amer. Math. Soc., Providence, RI, 2007.
    MathSciNet    

  7. L. Euler, De progressionibus harmonicis observationes, Comment. acad sci. Petrop. 7 (1740), 150-161. (Opera Omnia, Series 1, Vol. 14, 87-100.)

  8. S. S. Gelbert, E. M. Lapid and P. Sarnak, A new method for lower bounds of L-functions, C. R. Acad. Sci. Paris 339 (2004), 91-94.
    MathSciNet     CrossRef

  9. S. S. Gelbert, and F. Shahidi, Boundedness of automorphic L-functions in vertical strips, J. Amer. Math. Soc. 14 (2001), 79-107.
    MathSciNet     CrossRef

  10. T. Gillespie and G. Ji, Prime Number Theorems for Rankin-Selberg L-functions over number fields, Sci. China Math. 54 (2011), 35-46.
    MathSciNet     CrossRef

  11. T. Gillespie and Y. Ye, The Prime Number Theorem and Hypothesis H with lower order terms, J. Number Theory 141 (2014), 59-82.
    MathSciNet     CrossRef

  12. Y. Hashimoto, The Euler-Selberg constants for non-uniform lattices of rank one symmetric spaces, Kyushu J. Math. 57 (2003), 347-370.
    MathSciNet     CrossRef

  13. Y. Hashimoto, Y. Iijima, N. Kurokawa and M. Wakayama, Euler's constants for the Selberg and the Dedekind zeta functions, Bull. Belg. Math. Soc. Simon Stevin 11 (2004), 493-516.
    MathSciNet     CrossRef

  14. C. Hermite and T. J. Stieltjes, Correspondance d'Hermite et de Stieltjes, I & II, edited by B. Baillaud and H. Bourget, Gauthier-Villars, Paris, 1905.

  15. H. Jacquet and J. A. Shalika, On Euler products and the classification of automorphic representations I, Amer. J. Math. 103 (1981), 499-558.
    MathSciNet     CrossRef

  16. H. Jacquet and J. A. Shalika, On Euler products and the classification of automorphic representations II. Amer. J. Math. 103 (1981), 777-815.
    MathSciNet     CrossRef

  17. H. Kim, Functoriality for the exterior square of GL4 and the symmetric fourth of GL2, J. Amer. Math. Soc. 16 (2003), 139-183.
    MathSciNet     CrossRef

  18. H. Kim and F. Shahidi, Cuspidality of symmetric powers with applications, Duke Math. J. 112 (2002), 177-197.
    MathSciNet     CrossRef

  19. C. Knessl and M. W. Coffey, An asymptotic form for the Stieltjes constants γk(a) and for a sum Sγ(n) appearing under the Li criterion, Math. Comp. 80, No. 276 (2011), 2197-2217.
    MathSciNet     CrossRef

  20. X.-J. Li, The positivity of a sequence of numbers and the Riemann hypothesis, J. Number Theory 65 (1997), 325-333.
    MathSciNet     CrossRef

  21. J. Liu and Y. Ye, Perron's formula and the Prime Number Theorem for automorphic L-functions, Pure Appl. Math. Q. 3 (2007), 481-497.
    MathSciNet     CrossRef

  22. J. Liu and Y. Ye, Zeros of automorphic L-functions and noncyclic base change, in: Number Theory: Tradition and Modernization, Springer, New York, 2006, 119-152.
    MathSciNet    

  23. K. Maslanka, Li's criterion for the Riemann hypothesis - numerical approach, Opuscula Math. 24 (2004), 103-114.
    MathSciNet    

  24. C. Moeglin and J.-L. Waldspurger, Le spectre résiduel de GL(n), Ann. Sci. École Norm. Sup. 22 (1989), 605-674.
    MathSciNet     CrossRef

  25. C. J. Moreno, Explicit formulas in the theory of automorphic forms, in: Lecture Notes Math. Vol. 626, Springer, Berlin, 1977, 73-216.
    MathSciNet    

  26. M. R. Murty, Problems in Analytic Number Theory, Readings in Mathematics, GTM Springer-Verlag, 2001.
    MathSciNet    

  27. A. Odžak and L. Smajlović, On Li's coefficients for the Rankin-Selberg L-functions, Ramanujan J. 21 (2010), 303-334.
    MathSciNet     CrossRef

  28. Z. Rudnick and P. Sarnak, Zeros of principal L-functions and random matrix theory, Duke Math. J. 81 (1996), 269-322.
    MathSciNet     CrossRef

  29. P. Sarnak, Non-vanishing of L-functions on Re s = 1, in: Contributions to automorphic forms, geometry, and number theory, Johns Hopkins Univ. Press, Baltimore, 2004, 719-732.
    MathSciNet    

  30. A. Selberg, Old and new conjectures and results about a class of Dirichlet series, Collected papers, vol. II, Springer, 1991.
    MathSciNet    

  31. F. Shahidi, On nonvanishing of L-functions, Bull. Amer. Math. Soc. 2, No. 3 (1980), 462-464.
    MathSciNet     CrossRef

  32. F. Shahidi, On certain L-functions, Amer. J. Math. 103 (1981), 297-355.
    MathSciNet     CrossRef

  33. F. Shahidi, Fourier transforms of intertwinting operators and Plancherel measures for GL(n), Amer. J. Math. 106 (1984), 67-111.
    MathSciNet     CrossRef

  34. F. Shahidi, Local coefficients as Artin factors for real grups, Duke Math. J. 52 (1985), 973-1007.
    MathSciNet     CrossRef

  35. F. Shahidi, A proof of Langlands' conjecture on Plancherel measures, in: Complementary series for p-adic groups, Ann. Math. 132 (1990), 273-330.
    MathSciNet     CrossRef

  36. L. Smajlović, On Li's criterion for the Riemann hypotesis for the Selberg class, J. Number Theory 130 (2010), 828-851.
    MathSciNet     CrossRef

  37. E. C. Titchmarsh, The Theory of Functions, Second ed., Oxford University Press, Oxford, 1958.
    MathSciNet    

Glasnik Matematicki Home Page