Simple zeros on the critical line for weight two newforms of prime level, on harmonic average, without the Generalized Riemann Hypothesis

Let N be a prime and let H*(N) be the set of newforms of weight 2 and level N. For T > 0 and f ∈ H*(N) let N_f(T,2T) be the number of zeros β+iγ of L(s,f) with T < γ ≤ 2T, counted with multiplicity, and let N^s_{0,f}(T,2T) be the number of those that are simple and satisfy β = 1/2. Fix η, A > 0. We prove, without assuming any form of the Generalized Riemann Hypothesis, that uniformly for (log N)^η ≤ T ≤ (log N)^A, Σ_f ω_f N^s_{0,f}(T,2T) ≥ (C_MT − o(1)) Σ_f ω_f N_f(T,2T) as N → ∞, where ω_f are the harmonic (Petersson) weights and C_MT = 3/2 − (1/√2) cot(1/√2) = 0.6725… is the Montgomery–Taylor constant. For the number of distinct zeros, with no restriction on β, we obtain the constant (1 + C_MT)/2 = 0.8362…. The constant is the Montgomery–Taylor value of a bandwidth-one certificate; we do not attempt to improve it, and certificates of this shape are capped near 0.6818 in the zeta case. The proof transfers the rank–trace and inertia argument of Alpöge and Furman, in the form given by Hua and Yang for Dirichlet characters of prime modulus, to this family; that family averaging should restore bandwidth one was stated as a heuristic in an earlier manuscript. The family input is the Petersson large sieve with its exact diagonal term up to length N/(4π). At these heights every zero is controlled by a Gevrey window, so no zero-density estimate is needed. The result concerns a harmonically weighted family average; we do not obtain an unweighted proportion, nor any statement for an individual form.