Fractional moments of automorphic L-functions on GL(m)

Abstract

Let π be an irreducible unitary cuspidal representation of GL_m $\left( {\mathbb{A}_\mathbb{Q} } \right)$, m ≥ 2. Assume that π is self-contragredient. The author gets upper and lower bounds of the same order for fractional moments of automorphic L-function L(s, π) on the critical line under Generalized Ramanujan Conjecture; the upper bound being conditionally subject to the truth of Generalized Riemann Hypothesis.

Keywords

Moments / Automorphic L-functions / GRC

References

[1] Hardy G. H., Littlewood J. E.. Contributions to the theory of the Riemann zeta-function and the theory of the distribution of primes. Acta Math., 1918, 41: 119-196

[2] Ingham A. E.. Mean-value theorems in the theory of the Riemann zeta-function. Proc. London Math. Soc., 1926, 27(2): 273-300

[3] Ramacharadra K.. Some remarks on the mean value of the Riemann zeta-function and other Dirichlet series II. Hardy-Ramanujan J., 1980, 3: 1-25

[4] Ramacharadra K.. Some remarks on the mean value of the Riemann zeta-function and other Dirichlet series III. Ann. Acad. Sci. Fenn. Math., 1980, 5: 145-180

[5] Heath-Brown D. R.. Fractional moments of the Riemann zeta function. J. London Math. Soc., 1981, 24(2): 65-78

[6] Heath-Brown D. R.. Fractional moments of the Riemann zeta function (II). Quart. J. Math., 1993, 44: 185-197

[7] Soundararajan K.. Moments of the Riemann zeta-function. Ann. of Math., 2009, 170(2): 981-993

[8] Kačėnas A., Laurinčikas A., Zamarys S.. On fractional moments of Dirichlet L-function. Lith. Math. J., 2005, 45: 173-191

[9] Good A.. The mean square of Dirichlet series associated with cusp forms. Mathematika, 1982, 29: 278-295

[10] Laurinčikas A., Steuding J.. On fractional power moments of zeta-functions associated with certain cusp forms. Acta Appl. Math., 2007, 97: 25-39

[11] Sun H. W., Lü G. S.. On fractional power moments of L-functions associated with certain cusp forms. Acta Appl. Math., 2010, 109: 653-667

[12] Selberg A.. Old and new conjectures and results about a class of Dirichlet series, 1991, New York: Springer-Verlag 47-63

[13] Rudnick Z., Sarnak P.. Zeros of principal L-functions and random matrix theory. Duke Math. J., 1996, 81: 269-322

[14] Liu J. Y., Ye Y. B.. Selbergs orthogonality conjecture for automorphic L-functions. Amer. J. Math., 2005, 127: 837-849

[15] Deligne P.. La conjecture de Weil I. Inst. Hautes Études Sci. Publ. Math., 1974, 43: 273-307

[16] Fomenko O. M.. Fractional moments of automorphic L-functions. Algebra i Analiz, 2010, 22(2): 204-224 Translated edition, G. V. Kuzmina and O. M. Fomenko (eds.), St. Petersburg Math. J., 22(2), 2011, 321–335

[17] Wirsing E.. Das asymptotische Verhalten von Summen über multiplikative Funktionen. Math. Ann., 1961, 43: 75-102

[18] Wirsing E.. Das asymptotische Verhalten von Summen über multiplikative Funktionen II. Acta Math. Acad. Sci. Hungar., 1967, 18: 411-467

[19] Montgomery H. L., Vaughan R. C.. Hilbert inequality. J. London Math. Soc., 1974, 8: 73-82

[20] Gabriel R. M.. Some results concerning the integrals of moduli of regular functions along certain curves. J. London Math. Soc., 1927, 2: 112-117