On the density hypothesis for L-functions associated with holomorphic cusp forms
Revista Matemática Iberoamericana(2023)
摘要
We study the range of validity of the density hypothesis for the zeros of
L-functions associated with cusp Hecke eigenforms f of even integral weight
and prove that N_f(σ, T) ≪ T^2(1-σ)+ε holds for
σ≥ 1407/1601. This improves upon a result of Ivić, who had
previously shown the zero-density estimate in the narrower range σ≥
53/60. Our result relies on an improvement of the large value estimates for
Dirichlet polynomials based on mixed moment estimates for the Riemann zeta
function. The main ingredients in our proof are the Halász-Montgomery
inequality, Ivić's mixed moment bounds for the zeta function, Huxley's
subdivision argument, Bourgain's dichotomy approach, and Heath-Brown's bound
for double zeta sums.
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