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

Abstract

Let $\chi_m$ be a primitive real-valued Dirichlet character of conductor $m\ge3$, let $L(s)=L(s,\chi_m)$ be the associated Dirichlet $L$-function, and let $\xi_L(s)$ denote its completed form. For each $t \in \mathbb R,$ we define the entire function \[ H^L_t(z) := \int_{-\infty}^{\infty} e^{tu^2}\, \Phi_L(u)\, e^{izu}\, du, \] where $\Phi_L$ is the superexponentially decaying density \[ \Phi_L(u) := \frac{1}{2\pi}\int_{-\infty}^{\infty} \xi_L\!\left(\frac{1+is}{2}\right) e^{-isu}\, ds. \] The family $\{H_t^L\}_{t \in \mathbb R}$ describes the heat-flow evolution of $\xi_L$ and solves a backward heat equation. By the work of Newman, there exists a finite constant $\Lambda_L$, called the generalized de Bruijn--Newman constant, such that all the zeros of $H_t^L$ are real if and only if $t \ge \Lambda_L$. The Generalized Riemann Hypothesis (GRH) for $L$ is equivalent to the assertion that $\Lambda_L \le 0$. Dobner established the matching lower bound $\Lambda_L \ge 0$, while de Bruijn proved the upper bound $\Lambda_L \le \tfrac12$. We prove the unconditional uniform bound $\Lambda_L\le953/3200=0.2978125$ 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 $H_0^L$.