Isolated zeros of the Riemann zeta function and zeros of linear combinations of Dirichlet L-functions
Abstract
Let 0<γ1≤ γ2≤ ⋯ 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 ρn=βn+iγn is c-isolated if min(γn+1−γn, γn−γn−1)>2πc/log γn. Note that any isolated zero is necessarily simple and on the critical line. We prove unconditionally that there exists a positive constant c>0 such that at least 13/20 of the zeta zeros are c-isolated. Moreover, we can show that this proportion is more than 1/2 when c=1/6. Previously, it was not known unconditionally that there are infinitely many isolated zeros. 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 (which we extend to Dirichlet L-functions) 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 and have odd multiplicity. The previous unconditional record for this proportion was 7.38× 10−9, which was obtained by Dousselin who refined and made explicit a result of Selberg.
Related
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BibTeX
@misc{lamzouri2026isolated,
title = {{Isolated zeros of the Riemann zeta function and zeros of linear combinations of Dirichlet $L$-functions}},
author = {Youness Lamzouri},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26841},
eprint = {2610.26841},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26841},
}
RIS
TY - UNPB
TI - Isolated zeros of the Riemann zeta function and zeros of linear combinations of Dirichlet L-functions
AU - Youness Lamzouri
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.26841
UR - https://arxiv.org/abs/2610.26841
AB - Let 0<γ_1≤ γ_2≤ ⋯ 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 there exists a positive constant c>0 such that at least 13/20 of the zeta zeros are c-isolated. Moreover, we can show that this proportion is more than 1/2 when c=1/6. Previously, it was not known unconditionally that there are infinitely many isolated zeros. 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 (which we extend to Dirichlet L-functions) 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 and have odd multiplicity. The previous unconditional record for this proportion was 7.38× 10^−9, which was obtained by Dousselin who refined and made explicit a result of Selberg.
ER -