Isolated zeros of the Riemann zeta function and zeros of linear combinations of Dirichlet $L$-functions

Abstract

Let $0<γ_1\leq γ_2\leq \cdots$ be the ordinates of the non-trivial zeros of the Riemann zeta function in the upper half-plane, counted with multiplicity, and fix $c>0$. We say that a non-trivial zero $ρ_n=β_n+iγ_n$ is $c$-isolated if $\min(γ_{n+1}-γ_n, γ_n-γ_{n-1})>2πc/\log γ_n$. Note that any isolated zero is necessarily simple and on the critical line. We prove unconditionally that more than $50.87\%$ half of the non-trivial zeros are $1/6$-isolated. Previously, it was not known unconditionally that there are infinitely many isolated zeros. By allowing $c$ to be extremely small, we can push the above proportion to $13/20$. We also show that there are positive proportions of $k$ consecutive ``moderate'' gaps (gaps larger than a constant times the average local gap) between the ordinates of the zeta zeros, for $k=3, 4, 5$. As an application we combine our work on isolated zeros with an unconditional refinement of the Bombieri--Hejhal method to show unconditionally that at least $31\%$ of the zeros of a linear combination of two distinct primitive Dirichlet $L$-functions with the same conductor and parity lie on the critical line. The previous unconditional record for this proportion was $7.38\times 10^{-9}$, which was obtained by Dousselin who refined and made explicit a result of Selberg.