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 $L$-functions over short intervals
HTML articles powered by AMS MathViewer

by Guanghua Ji

Proc. Amer. Math. Soc. 137 (2009), 3569-3574

DOI: https://doi.org/10.1090/S0002-9939-09-10012-6

Published electronically: June 15, 2009

PDF | Request permission

Abstract:

Let $L(s,\pi )$ be the principal $L$-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 $L$-functions over short intervals.

References

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, DOI 10.1112/S0024611504015175
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, DOI 10.1090/S0894-0347-00-00351-9
Roger Godement and Hervé Jacquet, Zeta functions of simple algebras, Lecture Notes in Mathematics, Vol. 260, Springer-Verlag, Berlin-New York, 1972. MR 0342495
D. R. Heath-Brown, Fractional moments of the Riemann zeta function, J. London Math. Soc. (2) 24 (1981), no. 1, 65–78. MR 623671, DOI 10.1112/jlms/s2-24.1.65
Henryk Iwaniec and Emmanuel Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. MR 2061214, DOI 10.1090/coll/053
Hervé Jacquet and Joseph A. Shalika, A non-vanishing theorem for zeta functions of $\textrm {GL}_{n}$, Invent. Math. 38 (1976/77), no. 1, 1–16. MR 432596, DOI 10.1007/BF01390166
Jianya Liu and Yangbo Ye, Superposition of zeros of distinct $L$-functions, Forum Math. 14 (2002), no. 3, 419–455. MR 1899293, DOI 10.1515/form.2002.020
Wenzhi Luo, Zeév Rudnick, and Peter Sarnak, On the generalized Ramanujan conjecture for $\textrm {GL}(n)$, 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, DOI 10.1090/pspum/066.2/1703764
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, DOI 10.1215/S0012-7094-02-11114-4
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
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
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, DOI 10.1215/S0012-7094-96-08115-6
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