Zero density of L-functions related to Maass forms

Abstract

Let f(z) be a Hecke-Maass cusp form for SL₂(ℤ), and let L(s, f) be the corresponding automorphic L-function associated to f. For sufficiently large T, let N(σ, T) be the number of zeros ρ = β +iγ of L(s, f) with |γ| ⩽ T, β ⩾ σ, the zeros being counted according to multiplicity. In this paper, we get that for 3/4 ⩽ σ ⩽ 1 − ɛ, there exists a constant C = C(ɛ) such that N(σ,T) ≪ T^2(1−σ)/σ(logT) ^C, which improves the previous results.

Keywords

Maass form / automorphic L-function / zero density

References

[1] Heath-Brown D R. On the density of the zeros of the Dedekind zeta-function. Acta Arith, 1977, 33: 169-181

[2] Ivić A, Meurman T. Sums of coefficients of Hecke series. Acta Arith, 1994, LXVIII: 341-368

[3] Iwaniec H, Sarnak P. Perspectives of the analytic theory of L-functions. Geom Funct Anal, 2000 705 741

[4] Jutila M. Zero-density estimates for L-functions. Acta Arith, 1977, 32: 55-62

[5] Kim H H, Sarnak P. Refined estimates towards the Ramanujan and Selberg conjectures. J Amer Math Soc, 2003, 16: 175-181

[6] Kohnen W, Sankaranarayanan A, Sengupta J. The quadratic mean of automorphic L-functions. In: Automorphic Forms and Zeta Functions. Singapore: World Sci, 2005 262 279

[7] Lau Y K, Lü G S. Sums of Fourier coefficients of cusp forms. Quart J Math, 2011, 62: 687-716

[8] Sankaranarayanan A, Sengupta J. Zero-density estimate of L-functions attached to Maass forms. Acta Arith, 2007, 127: 273-284

[9] Xu Z. A new zero-density result of L-functions attached to Maass forms. Acta Math Sin (Engl Ser), 2011, 27: 1149-1162