REVIEW
Upper Bounds on the de Bruijn-Newman constant for Dirichlet $L$-functions with primitive real-valued characters
Number Theory · Submitted 19 Oct 2026 · 37 references
Pith's verdict: First non-trivial de Bruijn–Newman bounds for Dirichlet L-functions, with a solid analytic core and one decisive gap: 235 of 242 numerical certificates are private until the code is released.
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Referee report
The paper
The paper claims
For every primitive real Dirichlet character of conductor up to 400, the generalized de Bruijn–Newman constant is at most 953/3200 = 0.2978125 — the first nontrivial upper bound for any Dirichlet L-function.
The reading
What carries the argument
Three pieces carry the argument. The strip-contraction theorem [9] gives $\Lambda_L \le t_0 + y_0^2/2$ whenever $H^L_{t_0}$ has no zeros with $\operatorname{Im} z \ge y_0$, and the zero-dynamics machinery [7, 28] converts this into a velocity comparison: no zero of the moving family can cross a barrier above the strip, because the vertical velocity at the boundary is always more negative than the boundary's own rate of descent. The zero-free-region criterion (Lemma 3.3) reduces the bound to three explicitly described regions — an initial rectangle at $t = 0$ (covered for $x \le 2h_m$ by the GRH verification [25]), a moving barrier of width $\Delta_D = \sqrt{D^2 - y_0^2 - 2t_0}$ swept for $0 \le t \le t_0$, and an analytic right tail at the final time. The effective Riemann–Siegel approximation (Theorem 4.3) is the third piece: for $x \ge 480$, $|y| \le 1$, $0 \le t \le 1/2$, it writes $H^L_t(x+iy)/B^L_t(x+iy)$ as a pair of conductor-truncated Dirichlet sums plus an error $O_\le(e_A + e_B + e_{C,0})$ with every constant explicit. The conductor enters through the kernel $W_L(x) = \frac{\pi}{2m}\sum_{k=1}^{2m}\chi_m(k)e^{-i\pi k^2/m}\cot(\pi(x-k)/(2m))$, whose leading saddle coefficient $C^{(m)}_0(p) = e^{-i\pi/8}J_m(\delta)$ satisfies the uniform bound $|C^{(m)}_0(p)| \le B^\star_m < 3.33\sqrt{m}$ by a rearrangement argument that avoids incomplete character sums. The finitely many inequalities are finally checked in rigorous ball/interval arithmetic with a fixed numerical tolerance of $10^{-8}$.
Extended reading notes
Core claim
The central claim, Theorem 1.1, pairs a criterion with a computation. For each primitive real character $\chi_m$, attach the heat-flow family $H^L_t(z) = \int_{-\infty}^{\infty} e^{tu^2}\Phi_L(u)e^{izu}\,du$, whose zeros are all real exactly when $t \ge \Lambda_L$. If every zero of $H^L_0$ lies in the strip $|\operatorname{Im} z| \le D$, then $\Lambda_L \le \min\{D^2/2,\, \Lambda^\ast(m,\chi_m;D)\}$, where $\Lambda^\ast$ is the infimum of $t_0 + y_0^2/2$ over parameter triples whose three zero-free regions have been verified. At $D = 1$, the classical critical strip available unconditionally, the verified triples give $\Lambda_L \le 953/3200 = 0.2978125$ for all 242 real primitive characters with $3 \le m \le 400$, with individual bounds as low as 0.2278. The proof decomposes the required zero-free region into an initial rectangle at $t = 0$ (supplied by the published GRH verification [25]), a moving barrier swept out by the zero dynamics for $0 \le t \le t_0$, and an analytic right tail at the final time, and every error bound in the underlying effective Riemann–Siegel approximation is explicit and tracks the conductor $m$, uniform for $3 \le m \le 4\cdot10^5$.
Load-bearing premise
The unconditional bound rests on the correctness of the paper's private interval-arithmetic verification of regions (a)–(c) for the 242 characters — the code, mesh choices, and verification records are not public — and the conditional $D = 3/4$ rows additionally rest on the zero-free half-plane $\operatorname{Re}(s) > 7/8$ asserted in the unpublished preprint [24] rather than proved here.
Editorial extensions
If this is right
- Every one of the 242 real primitive characters with conductor up to 400 now has a verified bound $\Lambda_L \le 0.2978125 < 1/2$; combined with the non-negativity theorem of [11], each constant is bracketed in $[0, 0.2978125]$ rather than $[0, 1/2]$.
- The conditional gains are substantial: under the initial-strip hypothesis (for instance $|\operatorname{Im} z| \le 3/4$, which would follow from a zero-free half-plane $\operatorname{Re}(s) > 7/8$), 136 of the reference pairs and 149 cap-specific searches put $\Lambda_L$ strictly below $9/32 = 0.28125$.
- The approximation is strong enough to describe the zeros themselves: Theorem 8.1 gives a Riemann–von Mangoldt-type formula $N^L_t(X) = g_L(X,t) + O(\log(2+X))$ for $t > 0$, with all but finitely many large zeros real, simple, and spaced like $4\pi/\log(mX/4\pi)$.
- The paper's calibrated scale estimate (Remark 8.2) indicates that pushing the conductor-4 bound toward 0.1 would require a rigorous verification height on the order of $7\times10^{20}$, against the current $2.5\times10^7$; the logarithmic bottleneck is quantitative, not just qualitative.
Reading between the lines
- Until the certificate behind Theorem 1.1 is public — the paper states in Section 7.7 that its code, inputs, and verification records are retained in a private repository offered to referees — the unconditional bound is best read as a certified claim awaiting independent reproduction, and the decisive check is a re-implementation of regions (a)–(c) with different interval-arithmetic software.
- The $D = 3/4$ rows stand on the zero-free half-plane $\operatorname{Re}(s) > 7/8$ asserted in the unpublished preprint [24] rather than proved here; if that claim were withdrawn, the 149 improvements below $9/32$ would be lost and those characters would fall back to their unconditional $D = 1$ bounds.
- The analytic error bounds are uniform for conductors up to $4\cdot10^5$, but the verification height $h_m = 10^8/m$ shrinks with the conductor, so the practical ceiling of the method is the height of the rigorous GRH verification, not the quality of the approximation; raising that height (as was done for the zeta function up to $3\cdot10^{12}$) is the most direct path to stronger bounds.
- The architecture should transfer to any family with a functional equation and an effective Riemann–Siegel expansion with explicit conductor dependence, and the near-periodic large-zero spacing $4\pi/\log(mX/4\pi)$ predicted by Corollary 8.4 for $t > 0$ is a concrete pattern that numerical contouring could test without new theory.
Original 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 $H_0^L$.
The compiled record
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The abstract, sentence by sentence
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 $H_0^L$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | background 1.00 | 0.08 | 0.01 | asserted 0.80 |
| s1 | 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 \[ \Phi_L(u) := \frac{1}{2\pi}\int_{-\infty}^{\infty} \xi_L\!\left(\frac{1+is}{2}\right) e^{-isu}\, ds. | background 0.61 | 0.37 | 0.02 | asserted 0.97 |
| s2 | \] The family $\{H_t^L\}_{t \in \mathbb R}$ describes the heat-flow evolution of $\xi_L$ and solves a backward heat equation. | background 0.89 | 0.10 | 0.02 | asserted 1.00 |
| s3 | 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 $t \ge \Lambda_L$. | background 1.00 | 0.02 | 0.04 | asserted 1.00 |
| s4 | The Generalized Riemann Hypothesis (GRH) for $L$ is equivalent to the assertion that $\Lambda_L \le 0$. | background 0.98 | 0.04 | 0.03 | asserted 1.00 |
| s5 | Dobner established the matching lower bound $\Lambda_L \ge 0$, while de Bruijn proved the upper bound $\Lambda_L \le \tfrac12$. | background 0.98 | 0.03 | 0.03 | asserted 1.00 |
| s6 | 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. | result 0.94 | 0.97 | 0.95 | asserted 1.00 |
| s7 | The proof extends the effective approximation of Polymath to Dirichlet $L$-functions, retaining the conductor dependence in the error estimates. | method 0.92 | 0.75 | 0.03 | asserted 1.00 |
| s8 | We combine these estimates with Platt's numerical verification of GRH by Platt and rigorous interval computations. | method 0.99 | 0.76 | 0.04 | asserted 0.99 |
| s9 | We also give conditional bounds under explicit global initial zero-strip hypotheses, including $|\operatorname{Im}z|\le3/4$ for the zeros of $H_0^L$. | result 0.67 | 0.91 | 0.04 | asserted 1.00 |
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