REVIEW 3 major objections 4 minor 1 cited by
Pair correlation for Dedekind zeta functions of abelian extensions
T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read This paper proves a GRH-conditional pair-correlation formula for Dedekind zeta functions of abelian extensions and uses it to show that more than 45% of the zeros of quadratic-field zeta functions are distinct.
desk verdict A serious and probably correct extension of Montgomery's pair correlation to Dedekind zeta functions; the main soft spot is the compressed proof of uniformity near α=1, but the argument looks standard. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central object is the pair-correlation function $F_K(\alpha)$, defined as a normalized exponential sum over ordinates $\gamma,\gamma'$ with weight $w(u)=4/(4+u^2)$; Theorem 1.1 evaluates it by the classical pair-correlation method. The optimization machinery is the linear functional $Z_n(f)=n f(0)+2\int_0^1 f(x)x\,dx$ acting on even continuous $L^1$ functions $f$ with $\hat f(0)=1$, $\hat f\ge 0$, and $f(x)\le 0$ for $|x|\ge 1$. Semidefinite programming approximates the infimum of $Z_n(f)$, and the dimension-1 sphere-packing bound appears as the same optimization problem with the $n f(0)$ term replaced by $f(0)$ alone; the hat function $H(x)=\max(1-|x|,0)$ gives $Z_n(H)=n+1/3$, accounting for the final constant in Theorem 1.2.
What would settle it
Find or prove the existence of a zero of $\zeta_K(s)$ for an abelian $K$ with $\mathrm{Re}(s)\neq 1/2$; that alone refutes the conditional framework. Short of a GRH counterexample, compute $F_K(1)$ numerically for a fixed quadratic field at increasing $T$: the asserted uniformity near $\alpha=1$ predicts $F_K(1)=1+o(1)$, and a discrepancy that fails to shrink would locate the error in Lemma 3.1 and estimate (3.8).
Extended reading notes
Core claim
Under GRH, for an abelian number field $K$ of degree $n$, the pair correlation function of the zeros of $\zeta_K(s)$ has the explicit asymptotic $F_K(\alpha) = (n+o_K(1)) T^{-2|\alpha|}\log T + |\alpha| + o_K(1)$, uniformly for $|\alpha|\le 1$. The first term records the self-correlation at zero: each of the $n$ Dirichlet $L$-factors contributes a copy of the Riemann-zeta pair correlation, and the $|\alpha|$ term is the linear pair-correlation density familiar from random-matrix predictions. Feeding this into the linear functional $Z_n(f)=n f(0)+2\int_0^1 f(x)x\,dx$ over admissible Schwartz functions $f$ yields the multiplicity bound $N^*_K(T)\le (c_n+o_K(1))N_K(T)$ with $c_2=2.3226$, $c_3=3.3232$, $c_4=4.3235$, and $c_n\le (1+10^{-10})n+0.3243$; the hat function proves $c_n\le n+1/3$. From the quadratic bound and the known positive proportion of simple zeros, the paper derives $N_{K,d}(T)\ge (0.4585+o_K(1))N_K(T)$ for quadratic $K$, meaning more than 45% of the zeros are distinct.
Load-bearing premise
Everything rests on the Grand Riemann Hypothesis for Dedekind zeta functions: if a zero lies off the critical line, the pair-correlation formula, the constants $c_n$, and the 45% proportion are not established.
Editorial extensions
If this is right
- For every quadratic field, at least $(0.4585+o_K(1))N_K(T)$ of the first $N_K(T)$ zeros are distinct, an explicit proportion exceeding 45%.
- For cubic and quartic abelian fields the distinct-zero proportions are at least $0.2794$ and $0.1127$ respectively.
- The multiplicity counting function $N_K^*(T)$ grows at most like $2.3226N_K(T)$ for quadratic fields, $3.3232N_K(T)$ for cubic fields, and $4.3235N_K(T)$ for quartic fields.
- The pair-correlation formula gives a GRH-conditional spacing description for the zeros of all abelian-extensions' Dedekind zeta functions, not just for $\mathbb{Q}$.
- A single semidefinite program solves both the zeta-function pair-correlation bound and the dimension-1 sphere-packing bound, so numerical improvements to the optimizer transfer between the two problems.
Reading between the lines
- If the uniformity at $\alpha=1$ extends beyond the abelian splitting assumption, the same functional $Z_n$ could be applied to non-Galois extensions by replacing the Dirichlet-character factorization with a decomposition over irreducible Galois representations; the first test case would be a non-Galois cubic field.
- The bounded-support semidefinite formulation turns Fourier nonnegativity into a sum-of-squares condition, offering a template for certified bounds in other Fourier-constrained extremal problems, such as higher-dimensional sphere packing or counting problems with a fixed support.
- The constants suggest an additive term that stabilizes near $0.3243$ rather than $1/3$; if a future computation lowers the additive constant below $1/3$ for every $n$, that would show the extremizer is not the hat function and would indicate a genuinely different optimizer.
- The interpreted interpolation between pair correlation and sphere packing raises the possibility that the optimal functions for $Z_n$ are as hard to construct explicitly as the extremizers in dimensions 8 and 24, which would make the numerical constants the primary available description.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proves, conditionally on the Grand Riemann Hypothesis, an asymptotic formula for Montgomery's pair-correlation function F_K(alpha) for the zeros of the Dedekind zeta function of an abelian number field K of degree n. Theorem 1.1 states that F_K(alpha) = (n + o_K(1)) T^{-2|alpha|} log T + |alpha| + o_K(1) uniformly for |alpha| <= 1. Using this uniformity, the authors set up a semidefinite programming optimization over even Schwartz functions f satisfying f(x) <= 0 for |x| >= 1, fhat(0) = 1, and fhat >= 0, and obtain N_K^*(T) <= (c_n + o_K(1)) N_K(T) with c_2 = 2.3226, c_3 = 3.3232, c_4 = 4.3235, as well as general bounds n + 1/3 and (1 + 10^{-10})n + 0.3243. Corollary 1.3 gives that at least 45.85% of the zeros of quadratic fields, 27.94% for cubic fields, and 11.27% for quartic fields are distinct. The paper also discusses optimization over functions of bounded support as an interpolant between the Riemann-zeta pair-correlation problem and the one-dimensional Cohn-Elkies bound.
Significance. If the proof is completed, this is a substantial extension of Montgomery's pair-correlation method from the Riemann zeta function to Dedekind zeta functions of abelian extensions, with concrete numerical bounds that improve on what could be obtained by combining earlier results. A particular strength is that the numerical part is supported by reproducible code and interval-arithmetic verification, which makes the claimed constants checkable. The main intellectual contribution is the uniform version of Theorem 1.1 near alpha = 1 and its conversion, via Lemma 3.1, into a linear-functional optimization problem. The correctness of the final numerical theorems depends entirely on that uniformity, so the compressed treatment of the endpoint alpha = 1 is the central risk in the paper.
major comments (3)
- [Section 3.1, around Eq. (3.8)] The refined estimate (3.8) is the load-bearing uniformity needed for Lemma 3.1, because the integral in Lemma 3.1 runs over |alpha| <= 1 and therefore includes alpha = 1. The present text asserts (3.8) with the phrase 'using the same type of estimates as above' and an O(T) error uniformly for T^epsilon <= x <= T^2, but the derivation is not written out. In particular, the step S2 = O(x) is quoted from Goldston's thesis for unrestricted sums, whereas the sum here is over the character-restricted set S = {m : chi_i(m) = 1 for all i}. The authors should show that the conductor/discriminant dependence does not spoil the uniformity in x, and that the O(T) error is uniform up to x = T, i.e. up to alpha = 1. Without a complete argument, the conclusion 'each of the error terms is o(1) when epsilon <= alpha <= 1' is not established at the endpoint alpha = 1.
- [Section 2.1, Lemmas 2.1 and 2.2] Both Lemma 2.1 and Lemma 2.2 are central to the proof of Theorem 1.1, yet their proofs are either omitted or only outlined. Lemma 2.2 is used for every alpha, including the endpoint alpha = 1, and its error terms must be uniform in t after integration over [0,T]. The current text says the proof is 'a direct adaptation' of Montgomery's and that the term n log tau comes from the nonsymmetric functional equation, but it does not display the Dedekind-field calculation with r1 and r2. Since the paper's main new result depends on these estimates, the authors should either include complete proofs or give precise references that cover the Dedekind zeta case with explicit uniformity in the conductor.
- [Section 3.2, proof of the general bound in Theorem 1.2] The sentence explaining the bound c_n <= (1 + 10^{-10})n + 0.3243 says the values f(0) and Z_n(f) - n f(0) of a near-optimal function for n = 10^4 are used. Since Z_n(f) = n f(0) + 2 integral_0^1 f(x)x dx, a single feasible function can indeed give a bound for all n, but only if verified interval bounds on f(0) and on 2 integral_0^1 f(x)x dx are stated explicitly. As written, the passage from an optimization at n = 10^4 to an assertion for all n >= 1 is not fully transparent. The authors should spell out which single feasible function is being used and report the verified bounds on the two components of Z_n(f).
minor comments (4)
- [Throughout] There are several typographical errors that should be corrected: 'Theroem' in the Section 2.1 heading, 'satsifies' in Section 3.3, 'constriant' in Section 3.3, 'funcions' in Section 3.3, and 'is t hat is t' in Section 2.2.
- [Lemma 2.1 statement] In Lemma 2.1, the condition 's ≠ 1, 0, −m, ρ' is ambiguous; it should read 's not equal to 1, 0, a negative integer, or a zero ρ of ζ_K'.
- [Section 3.2, Eq. (3.9)] The passage from the distributional convergence T^{-2|alpha|} log T to δ_0 and the interchange with the o_K(1) error in F_K(alpha) deserves at least a footnote; as written it relies on continuity of f and uniform convergence on [-1,1], which is true, but the justification is only implicit.
- [Section 3.3] The statement that the bounded-support SDP recovers the best possible bound from [9] 'to within 70 decimals of accuracy' is striking, but the manuscript does not explain how this accuracy is measured or verified; a brief explanation would be helpful.
Circularity Check
No circularity: Theorem 1.1 is proved from explicit formulas and standard estimates; the SDP bounds are feasible-point upper bounds, not fitted to the conclusion.
full rationale
The paper's main derivation chain is self-contained in the relevant sense. Theorem 1.1 is obtained by adapting Montgomery's pair-correlation argument to Dedekind zeta functions: Lemma 2.1 and Lemma 2.2 are explicit-formula identities, and the subsequent estimates use standard PNT-in-arithmetic-progressions lemmas and Chebotarev/class field theory to identify the main term n log x. The uniformity near alpha = 1 follows by repeating Goldston's argument with the external thesis [18], not by importing the conclusion. Lemma 3.1 is a genuine inequality: any admissible f gives an upper bound N_K^*(T) <= (Z_n(f)+o(1))N_K(T), with the tail integral dropped using f <= 0 for |x| >= 1 and nonnegativity of F_K. The numerical constants are obtained by solving explicit semidefinite programs over feasible functions, with interval-arithmetic verification, rather than by optimizing against the desired 45 percent conclusion. The self-citation to [10] concerns the SDP modeling technique and the general optimization framework; no theorem from [10] is assumed in place of a proof. Benchmarks against known values from [25] and [9] are external checks. The compressed step (3.8) and the S2 = O(x) bound from Goldston's thesis are not written out in full, but this is a gap in exposition or proof detail, not circularity.
Assumptions & free parameters
free parameters (2)
- SDP degree d =
40 for the reported computations
- General bound slack epsilon =
10^-10
assumptions (5)
- domain assumption Grand Riemann Hypothesis for Dedekind zeta functions
- standard math Factorization of zeta_K into Dirichlet L-functions for abelian K
- standard math Chebotarev density theorem
- standard math Prime number theorem in arithmetic progressions and explicit formulas for psi(x, chi)
- standard math Paley-Wiener and Krein representation for nonnegative Fourier transforms with bounded support
Cite this review
Pith. "Pith review of Pair correlation for Dedekind zeta functions of abelian extensions." pith.science (2026). https://pith.science/paper/CA7CFK6Q
@misc{pith2026190804876,
author = {Pith},
title = {Pith review of: Pair correlation for Dedekind zeta functions of abelian extensions},
year = {2026},
howpublished = {\url{https://pith.science/paper/CA7CFK6Q}},
note = {Machine review of arXiv:1908.04876}
}
read the original abstract
Here we study problems related to the proportions of zeros, especially simple and distinct zeros on the critical line, of Dedekind zeta functions. We obtain new bounds on a counting function that measures the discrepancy of the zeta functions from having all zeros simple. In particular, for quadratic number fields, we deduce that more than 45% of the zeros are distinct. This extends work based on Montgomery's pair correlation approach for the Riemann zeta function. Our optimization problems can be interpreted as interpolants between the pair correlation bound for the Riemann zeta function and the Cohn-Elkies sphere packing bound in dimension 1. We compute the bounds through optimization over Schwartz functions using semidefinite programming and also show how semidefinite programming can be used to optimize over functions with bounded support.
Forward citations
Cited by 1 Pith paper
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Fourier optimization and pair correlation problems
For any sequence satisfying two axioms, long-term averages of its Montgomery-type form factor lie between explicit constants computed from reproducing kernels; the method refutes Gonek-Ki's conjecture for fixed c under RH.
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