REVIEW
An asymptotic Jensen-Polya criterion for GRH
Number Theory · Submitted 28 Oct 2026 · 14 references
Pith's verdict: A genuine new theorem — the first unshifted asymptotic Jensen–Pólya statement for all number fields — with a coherent saddle-point proof; the only real blemish is a Lean 4 certificate that is asserted but not checkable from the manuscript.
Why Pith says this
Highlight any text to comment on it.
Referee report
The paper
The paper claims
The paper proves that, for every number field, the unshifted Jensen polynomials of its Dedekind zeta function have, for all large degrees d, at least d − C_K√(d log d) distinct negative real zeros — the proportion of real roots tends to…
The reading
What carries the argument
The proof is a saddle-point analysis of the Cauchy integral representing the Jensen polynomial. Reciprocating and rescaling turns $J_{K,d}$ into a polynomial $A_d$ with the exact formula $A_d(x) = \frac{d!}{2\pi i d^d}\oint F_K(Bu)e^{dxu}u^{-d-1}\,du$, and setting $x=-q$ converts the sign of $A_d$ along the negative real axis into an oscillatory integral. The exponential phase is $h_q(u) = -qu + 2\kappa\sqrt{u} + \frac{2\kappa}{\Lambda}\sqrt{u}\log u - \log u$ with $\kappa = \pi/2$; it has two conjugate saddle points $u_q = r^2e^{-2i\alpha}$ once $q$ exceeds an edge value $q_e$, and their contributions combine into a positive multiple of a cosine. The upshot is the uniform oscillatory formula of Corollary 16, $A_d(-q) = 2\,\mathrm{Re}\,Z_d(q) + o(|Z_d(q)|)$, valid on a long interval $[q_e+\tau,\, c_0T]$, where the phase of $Z_d$ is $\vartheta_d(q) = dS(q) + O(1)$ and $S(q) = \mathrm{Im}\,h_q(u_q)$ is computed exactly, with $S'(q) = r^2\sin(2\alpha) > 0$. Since $S$ rises by nearly $\pi$ per unit of $q$, the phase crosses at least $\Delta_d/\pi$ multiples of $\pi$ and the cosine alternates sign that many times, with each sign change forcing a distinct real zero between crossings. The uniformity that makes the entire count work rests on two estimates: a concavity identity controlling the real part of the phase away from the saddles, and a complex Gaussian estimate (Lemma 14) that normalizes by $|D_q|$ rather than $\mathrm{Re}\,D_q$, so it survives the near-edge regime where the quadratic coefficient $D_q = u_q^2h''_q(u_q)$ is as small as $\sin^2\alpha$ and becomes almost purely oscillatory. Along the way the Dedekind zeta function is evaluated only in the half-plane $\mathrm{Re}\,s \ge 4$, where the Euler product converges absolutely, so no unproved input about its zeros enters.
Extended reading notes
Core claim
The central discovery, stated as Theorem 3, is that the unshifted Jensen polynomials of every Dedekind zeta function are asymptotically real-rooted: for each number field $K$ there are constants $C_K>0$ and $d_0(K)\ge 2$ such that for every integer $d\ge d_0(K)$, the polynomial $J_{K,d}(X)$ is nonzero and has at least $d - C_K\sqrt{d\log d}$ distinct negative real zeros. Equivalently, the proportion of real zeros is $1 - O_K(\sqrt{\log d/d})$, and the zeros that are counted lie in an explicit interval: in the variable of $J_{K,d}$ they occur for arguments contained in $[-C_1 d^{-1}(\log d)^2,\, -c_1/\log d]$. The proof is unconditional, assuming neither GRH nor $\zeta_K(1/2)\neq 0$, and the possibility that the central value vanishes — a phenomenon the paper illustrates with a cited example — is absorbed into the 'sufficiently large $d$' formulation, since $J_{K,d}$ is identically zero only below the order of vanishing of $F_K$ at zero. The authors are explicit about the boundary of the result: a growing number of nonreal zeros is still permitted, and the estimate yields neither a zero-free region nor a zero-density estimate, so it does not by itself decide GRH.
Load-bearing premise
The load-bearing premise is that every constant in the saddle-point chain — the concavity gap, the amplitude bounds, and above all the complex Gaussian approximation of Lemma 14 — is uniform in $d$ and $q$ across the whole counting interval $[q_e+\tau,\, c_0T]$, including the edge $q = q_e$, where the quadratic coefficient is as small as $\sin^2\alpha$, and the far end $q = c_0T$, where the Gaussian width parameter $n_q$ is only of order $\log d$; if any of those uniformities failed, the sign-change count that produces the zeros would collapse.
Editorial extensions
If this is right
- An asymptotic form of the Jensen–Pólya criterion now holds for every number field: the proportion of real zeros of the unshifted Jensen polynomials tends to one with quantified error $O_K(\sqrt{\log d/d})$, and for $K=\mathbb{Q}$ the bound $d - C\sqrt{d\log d}$ holds for every $d\ge 2$.
- The real zeros are located, not merely counted: they all lie in the explicit window $[-C_1 d^{-1}(\log d)^2,\, -c_1/\log d]$ in the variable of $J_{K,d}$, a logarithmic scale near zero (Remark 18).
- At most $C_K\sqrt{d\log d}$ of the $d$ roots escape the real axis, and even when the central value $\zeta_K(1/2)$ vanishes, $J_{K,d}$ is guaranteed nonzero for every $d\ge m_K$ and identically zero only below $m_K$.
- The authors state that the endpoint losses that produce the $\sqrt{d\log d}$ error 'capture the true difficulty of RH': any path from this theorem to full GRH must improve the control at the far endpoint $Q = c_0T$, where the Gaussian width parameter $n_q$ is only of order $\log d$.
- The paper also claims that the main theorems were autoformalized in the Lean 4 proof assistant, the Dedekind theorem assuming only that $K$ is a number field and neither GRH nor $\zeta_K(1/2)\neq 0$.
Reading between the lines
- Read as a quantitative measurement, the theorem shows where a full Jensen–Pólya proof of GRH would have to bite: the phase correction stays bounded precisely because $\zeta_K$ is evaluated in the region of absolute convergence, so pushing the interval further or shrinking the error would force the saddle amplitude to feel the zeta function inside the critical strip, the exact region the Riemann hy
- The cosine model predicts more than the theorem states: the real zeros should distribute nearly equidistantly in the phase $S(q)$, with the deficit $d - \#\{\text{real roots}\}$ scaling like $C_K\sqrt{d\log d}$; for $K=\mathbb{Q}$, where the coefficients are explicitly computable from the theta integral, a direct count of real roots over a range of degrees would test the uniformity of the Gaussian
- Because the machinery uses only the functional equation, the gamma factors, and Euler-product control in a fixed half-plane, the same trigonometric saddle-point strategy is a plausible template for other $L$-functions of the same completed-function shape, provided an analogue of the amplitude bound (Lemma 13) holds; the paper itself does not claim this extension.
- If the appended machine-checked certificate verifies, the paper would be an early instance of a new analytic number theory theorem published together with a proof assistant checking the proof itself; that would not alter the mathematics, but it would change the operational meaning of 'the theorem has been verified' for this style of asymptotic argument.
Original abstract
The Jensen--P\'olya criterion expresses the generalized Riemann hypothesis (GRH) for Dedekind zeta-functions as a real-rootedness condition on their Jensen polynomials. Previous asymptotic results require large shifts, while computations establish real-rootedness in the Riemann case through a finite range of degrees. Here we study the unshifted family as the degree tends to infinity. For every number field $K$, we prove unconditionally that $J_{K,d}(X)$ is nonzero and has at least $d-C_K\sqrt{d\log d}$ distinct negative real zeros for all sufficiently large $d$, where $C_K>0$ depends only on $K$. Therefore, the proportion of real zeros tends to one with error $O_K(\sqrt{\log d/d})$. The main theorems have been autoformalized in Lean~4 by AxiomProver.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
The Jensen--P\'olya criterion expresses the generalized Riemann hypothesis (GRH) for Dedekind zeta-functions as a real-rootedness condition on their Jensen polynomials. Previous asymptotic results require large shifts, while computations establish real-rootedness in the Riemann case through a finite range of degrees. Here we study the unshifted family as the degree tends to infinity. For every number field $K$, we prove unconditionally that $J_{K,d}(X)$ is nonzero and has at least $d-C_K\sqrt{d\log d}$ distinct negative real zeros for all sufficiently large $d$, where $C_K>0$ depends only on $K$. Therefore, the proportion of real zeros tends to one with error $O_K(\sqrt{\log d/d})$. The main theorems have been autoformalized in Lean~4 by AxiomProver.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The Jensen--P\'olya criterion expresses the generalized Riemann hypothesis (GRH) for Dedekind zeta-functions as a real-rootedness condition on their Jensen polynomials. | background 1.00 | 0.06 | 0.04 | asserted 1.00 |
| s1 | Previous asymptotic results require large shifts, while computations establish real-rootedness in the Riemann case through a finite range of degrees. | background 0.60 | 0.05 | 0.02 | asserted 1.00 |
| s2 | Here we study the unshifted family as the degree tends to infinity. | contribution 0.97 | 0.78 | 0.03 | asserted 0.70 |
| s3 | For every number field $K$, we prove unconditionally that $J_{K,d}(X)$ is nonzero and has at least $d-C_K\sqrt{d\log d}$ distinct negative real zeros for all sufficiently large $d$, where $C_K>0$ depends only on $K$. | result 0.99 | 0.95 | 0.89 | asserted 1.00 |
| s4 | Therefore, the proportion of real zeros tends to one with error $O_K(\sqrt{\log d/d})$. | result 0.56 | 0.90 | 0.26 | asserted 1.00 |
| s5 | The main theorems have been autoformalized in Lean~4 by AxiomProver. | method 0.56 | 0.47 | 0.03 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
References
Discussion
Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.