Zero-Density Concentration for Dirichlet Polynomials
Abstract
We prove that broad families of finite Dirichlet sums exhibit deterministic concentration of normalized vertical zero density on a single line. The main result gives a general criterion under which the normalized Jessen potentials converge locally uniformly to a piecewise-linear convex function with a single corner. The associated normalized measures describing the distribution of real parts of zeros in vertical mean density then converge weakly to a Dirac mass. Thus, for each fixed truncation, zeros may occupy a nontrivial range of real parts, while asymptotically their normalized vertical density concentrates on one line. The proof combines the finite Bohr lift with a translation-uniform anti-concentration estimate for isolated prime coordinates. We apply the criterion to partial sums of the Riemann zeta function, fixed Dirichlet L-functions, and primitive holomorphic Hecke eigenforms of fixed level, trivial Dirichlet character, and without complex multiplication. The concentration line is Re s=1/2 in the normalized setting and becomes Re s=k/2 for the classical Fourier coefficients of a form of weight k.
Related
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BibTeX
@misc{dubon2026zerodensity,
title = {{Zero-Density Concentration for Dirichlet Polynomials}},
author = {Eric Dubon},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17875},
eprint = {2609.17875},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17875},
}
RIS
TY - UNPB
TI - Zero-Density Concentration for Dirichlet Polynomials
AU - Eric Dubon
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.17875
UR - https://arxiv.org/abs/2609.17875
AB - We prove that broad families of finite Dirichlet sums exhibit deterministic concentration of normalized vertical zero density on a single line. The main result gives a general criterion under which the normalized Jessen potentials converge locally uniformly to a piecewise-linear convex function with a single corner. The associated normalized measures describing the distribution of real parts of zeros in vertical mean density then converge weakly to a Dirac mass. Thus, for each fixed truncation, zeros may occupy a nontrivial range of real parts, while asymptotically their normalized vertical density concentrates on one line. The proof combines the finite Bohr lift with a translation-uniform anti-concentration estimate for isolated prime coordinates. We apply the criterion to partial sums of the Riemann zeta function, fixed Dirichlet L-functions, and primitive holomorphic Hecke eigenforms of fixed level, trivial Dirichlet character, and without complex multiplication. The concentration line is Re s=1/2 in the normalized setting and becomes Re s=k/2 for the classical Fourier coefficients of a form of weight k.
ER -