On the density hypothesis for $L$-functions associated with holomorphic cusp forms

Abstract

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 $N_f(\sigma ,T)\ll T^{2(1-\sigma )+\epsilon }$ holds for $\sigma \ge 1407/1601$. This improves upon a result of Ivić, who had previously shown the zero-density estimate in the narrower range $\sigma \ge 53/60$. 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.