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Selberg's Normal Density Theorem for Automorphic L-Functions for GL m

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Abstract

Let π be an irreducible unitary cuspidal representation of GL m \({(\Bbb {A}_\Bbb{Q})}\) with m ≥ 2, and L(s, π) the L-function attached to π. Under the Generalized Riemann Hypothesis for L(s, π), we estimate the normal density of primes in short intervals for the automorphic L-function L(s, π). Our result generalizes the corresponding theorem of Selberg for the Riemann zeta-function.

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References

  1. Godement, R., Jacquet, H.: Zeta functions of simple algebras, Lecture Notes in Math., 260, Springer-Verlag, Berlin, 1972

  2. Jacquet, H., Shalika, J. A.: On Euler products and the classification of automorphic representations I. Amer. J. Math., 103, 499–558 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  3. Liu, J. Y., Ye, Y. B.: Perron's formula and the prime number theorem for automorphic L-functions. Pure Appl. Math. Quart., to appear

  4. Iwaniec, H., Kowalski, E.: Analytic Number Theory, Amer. Math. Soc. Colloquium Publ., 53, Amer. Math. Soc., Providence, 2004

  5. Qu, Y.: The prime number theorem for automorphic L-functions for GL m , J. Number Theory, 122, 84–89 (2007)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  7. Selberg, A.: On the normal sensity of primes in short intervals and the difference between consecutive primes. Archiv f. Math. og Naturvid. B, 47(6), (1943)

  8. Deligne, P.: La conjecture de Weil, I. Inst. Hautes Ètudes Sci. Publ. Math., 43, 206–226 (1972)

    MathSciNet  Google Scholar 

  9. Gallagher, P. X.: Some consequences of the Riemann hypothesis. Acta. Arith., 37, 339–343 (1980)

    MATH  MathSciNet  Google Scholar 

  10. Gallagher, P. X.: A large sieve density estimate near σ = 1. Invent. Math., 11, 329–339 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  11. Liu, J. Y., Ye, Y. B.: Superpositions of distinct L-functions. Forum Math., 14, 419–455 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  12. Shahidi, F.: On certain L-functions, Amer. J. Math., 103, 297–355 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  13. Shahidi, F.: Fourier transforms of intertwining operators and Plancherel measures for GL(n). Amer. J. Math., 106, 67–111 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  14. Shahidi, F.: Local coefficients as Artin factors for real groups, Duke Math. J., 52, 973–1007 (1985)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  16. Gelbart, S. S., Shahidi, F.: Boundedness of automorphic L-functions in vertical strips. J. Amer. Math. Soc., 14, 79–107 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  17. Moreno, C. J.: Explicit formulas in the theory of automorphic forms, Lecture Notes Math., Springer, Berlin, 626, 730–216, 1977

  18. Moreno, C. J.: Analytic proof of the strong multiplicity one theorem. Amer. J. Math., 107, 163–206 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  19. Jacquet, H., Shalika, J. A.: On Euler products and the classification of automorphic representations II. Amer. J. Math., 103, 777–815 (1981)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Shandong University, Jinan, 250100, P. R. China

    Yan Qu

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  1. Yan Qu
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Correspondence to Yan Qu.

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Supported by NSFC Grant #10531060, and by a Ministry of Education Major Grant Program in Sciences and Technology

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Qu, Y. Selberg's Normal Density Theorem for Automorphic L-Functions for GL m . Acta Math Sinica 23, 1903–1908 (2007). https://doi.org/10.1007/s10114-005-0926-5

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  • DOI: https://doi.org/10.1007/s10114-005-0926-5

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