Lower bounds for moments of automorphic 𝐿-functions over short intervals

Ji, Guanghua

doi:10.1090/S0002-9939-09-10012-6

Lower bounds for moments of automorphic -functions over short intervals

Author: Guanghua Ji
Journal: Proc. Amer. Math. Soc. 137 (2009), 3569-3574
MSC (2000): Primary 11F66, 11M26, 11M41
DOI: https://doi.org/10.1090/S0002-9939-09-10012-6
Published electronically: June 15, 2009
MathSciNet review: 2529862
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Abstract: Let $L(s,\pi)$ be the principal -function attached to an irreducible unitary cuspidal automorphic representation $\pi$ of $GL_m(\mathbb{A}_\mathbb{Q})$ . The aim of the paper is to give a simple method to show the lower bounds of mean value for automorphic -functions over short intervals.

1. J. B. Conrey, D. W. Farmer, J. P. Keating, M. O. Rubinstein, and N. C. Snaith, Integral moments of 𝐿-functions, Proc. London Math. Soc. (3) 91 (2005), no. 1, 33–104. MR 2149530, https://doi.org/10.1112/S0024611504015175
2. Stephen Gelbart and Freydoon Shahidi, Boundedness of automorphic 𝐿-functions in vertical strips, J. Amer. Math. Soc. 14 (2001), no. 1, 79–107. MR 1800349, https://doi.org/10.1090/S0894-0347-00-00351-9
3. Roger Godement and Hervé Jacquet, Zeta functions of simple algebras, Lecture Notes in Mathematics, Vol. 260, Springer-Verlag, Berlin-New York, 1972. MR 0342495
4. D. R. Heath-Brown, Fractional moments of the Riemann zeta function, J. London Math. Soc. (2) 24 (1981), no. 1, 65–78. MR 623671, https://doi.org/10.1112/jlms/s2-24.1.65
5. Henryk Iwaniec and Emmanuel Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. MR 2061214
6. Hervé Jacquet and Joseph A. Shalika, A non-vanishing theorem for zeta functions of 𝐺𝐿_{𝑛}, Invent. Math. 38 (1976/77), no. 1, 1–16. MR 432596, https://doi.org/10.1007/BF01390166
7. Jianya Liu and Yangbo Ye, Superposition of zeros of distinct 𝐿-functions, Forum Math. 14 (2002), no. 3, 419–455. MR 1899293, https://doi.org/10.1515/form.2002.020
8. Wenzhi Luo, Zeév Rudnick, and Peter Sarnak, On the generalized Ramanujan conjecture for 𝐺𝐿(𝑛), Automorphic forms, automorphic representations, and arithmetic (Fort Worth, TX, 1996) Proc. Sympos. Pure Math., vol. 66, Amer. Math. Soc., Providence, RI, 1999, pp. 301–310. MR 1703764
9. Giuseppe Molteni, Upper and lower bounds at 𝑠=1 for certain Dirichlet series with Euler product, Duke Math. J. 111 (2002), no. 1, 133–158. MR 1876443, https://doi.org/10.1215/S0012-7094-02-11114-4
10. K. Ramachandra, Some remarks on the mean value of the Riemann zeta function and other Dirichlet series. I, Hardy-Ramanujan J. 1 (1978), 15. MR 565298
11. K. Ramachandra, Some remarks on the mean value of the Riemann zeta function and other Dirichlet series. I, Hardy-Ramanujan J. 1 (1978), 15. MR 565298
12. Zeév Rudnick and Peter Sarnak, Zeros of principal 𝐿-functions and random matrix theory, Duke Math. J. 81 (1996), no. 2, 269–322. A celebration of John F. Nash, Jr. MR 1395406, https://doi.org/10.1215/S0012-7094-96-08115-6
13. E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1986. Edited and with a preface by D. R. Heath-Brown. MR 882550