Upper Bounds on the de Bruijn-Newman constant for Dirichlet $L$-functions with primitive real-valued characters

Source abstract

Let $χ_m$ be a primitive real-valued Dirichlet character of conductor $m\ge3$, let $L(s)=L(s,χ_m)$ be the associated Dirichlet $L$-function, and let $ξ_L(s)$ denote its completed form. For each $t \in \mathbb R,$ we define the entire function $$H^L_t(z) := \int_{-\infty}^{\infty} e^{tu^2}\, Φ_L(u)\, e^{izu}\, du,$$ where $Φ_L$ is the superexponentially decaying density $$Φ_L(u) := \frac{1}{2π}\int_{-\infty}^{\infty} ξ_L\!\left(\frac{1+is}{2}\right) e^{-isu}\, ds.$$ The family $\{H_t^L\}_{t \in \mathbb R}$ 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 $t \ge Λ_L$. The Generalized Riemann Hypothesis (GRH) for $L$ is equivalent to the assertion that $Λ_L \le 0$. Dobner established the matching lower bound $Λ_L \ge 0$, while de Bruijn proved the upper bound $Λ_L \le \tfrac12$. We prove the unconditional uniform bound $Λ_L\le953/3200=0.2978125$ for every primitive real-valued character of conductor $3\le m\le400$, 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 $|\operatorname{Im}z|\le3/4$ for the zeros of $H_0^L$.