Simple zeros on the critical line of primitive Dirichlet L-functions on average over moduli q ≤ Q: the killed kernel and the Shell kernel

Abstract

Let F_Q be the set of primitive Dirichlet characters of modulus 1 < q ≤ Q and T = (log Q)^(r₀ + ε) with r₀ ≥ 3, ε > 0 fixed. The preprint [9] proved that, as Q → ∞, at least 0.7212 of the zeros of the L(s, χ), χ ∈ F_Q, with ordinate in (T, 2T], counted with multiplicity and summed over the family, are simple and lie on the critical line. We raise this constant in two ways. First, a large-sieve lemma for the part of the dual variable beyond the diagonal (Farey points of level Q avoid the major arcs of small denominator) gives the constant 0.7235 with the rate log log Q/log Q and no hypothesis of any kind. Second, the “Shell route”, which uses the explicit formula for the characters of small modulus and two published zero-density estimates for the family (Jutila's log-free hybrid estimate and Montgomery's hybrid estimate), gives 0.9059137927, with an error O((log Q)^(−θ)) for every θ < 503/1994. Both results are asymptotic; the thresholds implicit in the proofs are very large. Each is proved on paper, and each statement is a Lean 4 theorem in the release of the accompanying Lean repository that depends only on the three standard axioms: the first has no hypothesis beyond r₀ ≥ 3, ε > 0, and the second none beyond the two zero-density estimates and r₀ ≥ 3, ε > 0, 0 < θ < 503/1994 (§1.5). A corresponding value for the moduli Q/2 < q ≤ Q is not claimed. The results are family averages and say nothing about an individual L-function.

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