REVIEW
The Riemann zeta function at the zeros of real-rooted trigonometric polynomials
Number Theory · Submitted 19 Oct 2026 · 21 references
Pith's verdict: A genuinely new extension of Li-Radziwill using crystalline measures; Theorem 1 looks solid, Theorem 2 is plausible but compressed, and the paper deserves careful refereeing.
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Referee report
The paper
The paper claims
For any trigonometric polynomial whose zeros are all real, at most two-thirds of its zeros in each dyadic interval [T,2T] can be ordinates of Riemann zeros, and |ζ(1/2+iγ)| reaches at least exp((1+o(1))√(log T/(6 log log T))) at one of…
The reading
What carries the argument
The load-bearing object is the Kurasov–Sarnak / Alon–Cohen–Vinzant Fourier summation formula (Theorem 4): for every Schwartz function $\varphi$, $\sum_{P(\gamma)=0}\varphi(\gamma) = c_0\widehat{\varphi}(0) + \sum_{a\in\mathbb{Z}_+^M} c_a\widehat{\varphi}(\ell\cdot a)$, with $c_a = -(a\cdot\ell)[z^a]\log(L(z)/L(0))$, $c_a = O(|a|)$, and $\sum_{|\ell\cdot a|\le A}|c_a| = O(A^{M+1})$. It is the bridge between the zero set of $P$ and the analytic arguments: Sections 3 and 4 feed it with the mollified zeta weights $\varphi(x)=\phi(x/T)$ and $s^{-ix}\phi(x/T)$, and Section 6 feeds it with the resonator, so that in every case a discrete average over $P$'s zeros is identified with its Fourier-side continuous integral up to acceptable error.
Extended reading notes
Core claim
The paper's central claim is that the two theorems previously proved only for vertical arithmetic progressions hold verbatim for every real-rooted trigonometric polynomial, with no arithmetic-structure hypothesis on its zero set. The transport is one object: the Kurasov–Sarnak Fourier summation formula, which asserts that for every Schwartz test function $\varphi$, the zero-counting sum $\sum_{P(\gamma)=0}\varphi(\gamma)$ equals $c_0\widehat{\varphi}(0) + \sum_{a\in\mathbb{Z}_+^M} c_a\widehat{\varphi}(\ell\cdot a)$, with coefficients $c_a = -(a\cdot\ell)[z^a]\log(L(z)/L(0))$ of size $O(|a|)$. This formula replaces Poisson summation and converts every discrete average over $P$'s zeros into a continuous integral, the error being controlled by the polynomial growth of the coefficients. Proposition 5 shows the discrete mollified first and second moments of $\zeta$ agree with their continuous counterparts up to $o(T)$; combined with the second-moment bound of Balasubramanian–Conrey–Heath-Brown and Cauchy–Schwarz, that forces at least one third of $P$'s zeros to be free of zeta zeros, giving Theorem 1. For Theorem 2, a Soundararajan-type resonator satisfies the expected sample-size conditions (Lemma 10), the genuinely new step being the removal of a thin set of exceptional primes so that crystalline frequencies lying extremely close to logarithms of rationals cannot spoil the main term.
Load-bearing premise
The argument stands on the imported Kurasov–Sarnak summation formula (Theorem 4), together with the companion count $\#\{\gamma\in[T,2T]:P(\gamma)=0\}=c_0T(1+o(1))$: if the formula held only for a restricted class of test functions, or if the coefficient bounds $c_a=O(|a|)$ and the polynomial mass-growth failed, the moment comparisons behind Theorem 1 and the resonator estimates behind Theorem 2 would break.
Editorial extensions
If this is right
- Theorem 1 makes the two-thirds non-coincidence bound unconditional for every real-rooted trigonometric polynomial, with no arithmetic-structure hypothesis on the zero set; aperiodic zero sets are covered.
- Theorem 2 reproduces the Li–Radziwill large-value size at P's own zeros: in every dyadic interval [T,2T], |ζ(1/2+iγ)| exceeds exp((1+o(1))√(log T/(6 log log T))) at some zero γ of P.
- Corollary 3 carries the two-thirds bound to the Kirchhoff–Neumann spectrum: at most two-thirds of the spectral values of a finite compact metric graph in each dyadic interval are zeta-zero ordinates, counting multiplicity, in the limsup sense.
- Proposition 5, the equality of the discrete and continuous mollified moments up to o(T), is stated to be of independent interest for later discrete sampling questions.
Reading between the lines
- The persistence of the exact constants (2/3 and √(1/6)) from arithmetic progressions to arbitrary real-rooted zero sets suggests the bounds are set by the mollification and resonance framework itself, not by the placement of P's zeros; one would not expect different constants for different real-rooted P.
- Because the only structural input is a Fourier summation formula with polynomial coefficient growth, the machinery should port to any other zero set (for example, the spectra of quantum graphs, already touched by Corollary 3) whose counting measure admits such a formula.
- If the authors' stated belief that the results hold for any reasonable trigonometric polynomial, real-rooted or not, is correct, then the crystalline summation formula is the critical missing ingredient, since non-real-rooted polynomials presently lack an explicit summation formula of this type.
Original abstract
Let $P$ be a trigonometric polynomial with only real zeros. We prove that at most $\frac{2}{3}$ of its zeros coincide with the zeros of the Riemann zeta function $\zeta(s)$ on the critical line. We also show that \begin{equation*} \sup_{\substack{T\leq x\leq2T\\ P(x)=0}} \log \left|\zeta\left(\frac{1}{2}+ix\right)\right| \geq (1+o(1))\sqrt{\frac{\log T}{6\log\log T}}. \end{equation*} These results extend the corresponding theorems of Li and Radziwill on the zeros of $\zeta(s)$ along vertical arithmetic progressions. The main novelty in our approach is the use of the work of Kurasov and Sarnak and Lee-Yang representations of Alon-Cohen-Vinzant, which show that the corresponding measure $\sum_{P(\gamma)=0} {\bf \delta}_\gamma$ is crystalline.
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
Let $P$ be a trigonometric polynomial with only real zeros. We prove that at most $\frac{2}{3}$ of its zeros coincide with the zeros of the Riemann zeta function $\zeta(s)$ on the critical line. We also show that \begin{equation*} \sup_{\substack{T\leq x\leq2T\\ P(x)=0}} \log \left|\zeta\left(\frac{1}{2}+ix\right)\right| \geq (1+o(1))\sqrt{\frac{\log T}{6\log\log T}}. \end{equation*} These results extend the corresponding theorems of Li and Radziwill on the zeros of $\zeta(s)$ along vertical arithmetic progressions. The main novelty in our approach is the use of the work of Kurasov and Sarnak and Lee-Yang representations of Alon-Cohen-Vinzant, which show that the corresponding measure $\sum_{P(\gamma)=0} {\bf \delta}_\gamma$ is crystalline.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $P$ be a trigonometric polynomial with only real zeros. | background 0.92 | 0.09 | 0.02 | asserted 0.81 |
| s1 | We prove that at most $\frac{2}{3}$ of its zeros coincide with the zeros of the Riemann zeta function $\zeta(s)$ on the critical line. | result 1.00 | 0.96 | 0.69 | asserted 1.00 |
| s2 | We also show that \begin{equation*} \sup_{\substack{T\leq x\leq2T\\ P(x)=0}} \log \left|\zeta\left(\frac{1}{2}+ix\right)\right| \geq (1+o(1))\sqrt{\frac{\log T}{6\log\log T}}. | result 1.00 | 0.94 | 0.18 | asserted 1.00 |
| s3 | \end{equation*} These results extend the corresponding theorems of Li and Radziwill on the zeros of $\zeta(s)$ along vertical arithmetic progressions. | interpretation 0.60 | 0.23 | 0.05 | asserted 0.98 |
| s4 | The main novelty in our approach is the use of the work of Kurasov and Sarnak and Lee-Yang representations of Alon-Cohen-Vinzant, which show that the corresponding measure $\sum_{P(\gamma)=0} {\bf \delta}_\gamma$ is crystalline. | contribution 0.68 | 0.29 | 0.04 | asserted 1.00 |
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References
Discussion
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