Upper Bounds on the de Bruijn-Newman constant for Dirichlet L-functions with primitive real-valued characters
Abstract
Let χm be a primitive real-valued Dirichlet character of conductor m≥3, let L(s)=L(s,χm) be the associated Dirichlet L-function, and let ξL(s) denote its completed form. For each t ∈ ℝ, we define the entire function HLt(z) := ∫−∞∞ etu2 ΦL(u) eizu du, where ΦL is the superexponentially decaying density ΦL(u) := 1/2π∫−∞∞ ξL ((1+is)/2) e−isu ds. The family {HtL}t ∈ ℝ describes the heat-flow evolution of ξL and solves a backward heat equation. By the work of Newman, there exists a finite constant ΛL, called the generalized de Bruijn--Newman constant, such that all the zeros of HtL are real if and only if t ≥ ΛL. The Generalized Riemann Hypothesis (GRH) for L is equivalent to the assertion that ΛL ≤ 0. Dobner established the matching lower bound ΛL ≥ 0, while de Bruijn proved the upper bound ΛL ≤ 1/2. We prove the unconditional uniform bound ΛL≤953/3200=0.2978125 for every primitive real-valued character of conductor 3≤ m≤400, together with stronger individual bounds. The proof extends the effective approximation of Polymath to Dirichlet L-functions, retaining the conductor dependence in the error estimates. We combine these estimates with Platt's numerical verification of GRH by Platt and rigorous interval computations. We also give conditional bounds under explicit global initial zero-strip hypotheses, including |Imz|≤3/4 for the zeros of H0L.
Related
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- Irrationality of values of L-functions for real Dirichlet characters
- Hypothetical connection of the theta functions of Dirichlet characters with the real cyclotomic fields
- Improved first moments of primitive quartic and sextic Dirichlet L-functions
- Angular Monotonicity of Completed Dirichlet L-Functions via the Three-Term Zeta Decomposition: A Proof of the Generalized Riemann Hypothesis for Non-Principal Characters
BibTeX
@misc{freire2026upper,
title = {{Upper Bounds on the de Bruijn-Newman constant for Dirichlet $L$-functions with primitive real-valued characters}},
author = {Ricardo Freire and Cristian González-Riquelme and Claudio Muñoz and Felipe Poblete and Vicente Salinas},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27343},
eprint = {2610.27343},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27343},
}
RIS
TY - UNPB
TI - Upper Bounds on the de Bruijn-Newman constant for Dirichlet L-functions with primitive real-valued characters
AU - Ricardo Freire
AU - Cristian González-Riquelme
AU - Claudio Muñoz
AU - Felipe Poblete
AU - Vicente Salinas
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.27343
UR - https://arxiv.org/abs/2610.27343
AB - Let χ_m be a primitive real-valued Dirichlet character of conductor m≥3, let L(s)=L(s,χ_m) be the associated Dirichlet L-function, and let ξ_L(s) denote its completed form. For each t ∈ ℝ, we define the entire function H^L_t(z) := ∫_−∞^∞ e^tu^2 Φ_L(u) e^izu du, where Φ_L is the superexponentially decaying density Φ_L(u) := 1/2π∫_−∞^∞ ξ_L ((1+is)/2) e^−isu ds. The family {H_t^L}_t ∈ ℝ describes the heat-flow evolution of ξ_L and solves a backward heat equation. By the work of Newman, there exists a finite constant Λ_L, called the generalized de Bruijn--Newman constant, such that all the zeros of H_t^L are real if and only if t ≥ Λ_L. The Generalized Riemann Hypothesis (GRH) for L is equivalent to the assertion that Λ_L ≤ 0. Dobner established the matching lower bound Λ_L ≥ 0, while de Bruijn proved the upper bound Λ_L ≤ 1/2. We prove the unconditional uniform bound Λ_L≤953/3200=0.2978125 for every primitive real-valued character of conductor 3≤ m≤400, together with stronger individual bounds. The proof extends the effective approximation of Polymath to Dirichlet L-functions, retaining the conductor dependence in the error estimates. We combine these estimates with Platt's numerical verification of GRH by Platt and rigorous interval computations. We also give conditional bounds under explicit global initial zero-strip hypotheses, including |Imz|≤3/4 for the zeros of H_0^L.
ER -