{"id":"e01d4bd4-e2c4-4abc-8946-1f7b7fb63dd3","arxiv_id":"2610.26705","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"For any real-rooted trigonometric polynomial P, at most 2/3 of its zeros in [T,2T] can be critical-line zeros of zeta, and |zeta(1/2+it)| must exceed exp((1+o(1)) sqrt(log T / (6 log log T))) at some zero of P.","lead":"At most two thirds of the zeros of any real-rooted trigonometric polynomial can also be zeros of the Riemann zeta function on the critical line, and the zeta function must still reach very large values somewhere on such a zero set. The proof extends results previously known for arithmetic progressions by importing \"crystalline measure\" technology into analytic number theory.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek/deepseek-v4-flash","headline":"Theorem 2 depends on Lemma 10's asserted inequality (20); the proof does not show how negative crystalline frequencies with a/b much less than 1 are covered, so the high-frequency error bound is not established.","rationale":"The reader's verdict is already CONDITIONAL and flags Theorem 4 and Lemma 9 as fragile imported inputs; my stress test points instead at Lemma 10's inequality (20), which is the least secure internal step in Theorem 2. It is not a demonstrated error: the inequality may be provable with a longer argument, for instance by using the distance condition to force b/a to be controlled in the relevant range or by a more careful S_X embedding. But as written the proof gives no indication of how negative frequencies are handled, and every displayed bound leading to (21) is asserted without derivation. This is precisely the kind of step that should be checked before the paper is accepted, and the paper's own \"brief comment\" signals the gap. Since the concern is a missing verification rather than a known counterexample, I recommend no change to the reader's CONDITIONAL verdict. I partially agree with the reader: the imported summation formula and Lemma 9 are also load-bearing, but the most serious unverified internal estimate is the high-frequency control in Lemma 10.","tokens_in":22859,"tokens_out":51771,"duration_ms":569639,"concrete_test":"Verify inequality (20) by writing out the full parametrization with the nh <= H restriction: for each a in Z_+^M with |l dot a| > R, including those with l dot a < 0, show that every admissible (n1,k,u) maps injectively (or with controlled multiplicity) to a term of S_X with product <= X. A concrete computational check is to instantiate the resonator for M = 1, l = 1, eta = 1/240, T = 10^6 (so X = T^{1/6-20eta}, H = T^{1/2-eta}) and evaluate U(a,b), r(a)K(b), and S_X for a = 1 and b a prime between L^2 and exp((log L)^2); if U > r(a)K(b)S_X for any such pair with b <= H, Lemma 10 as stated fails. For the analytic direction, re-derive (21), especially the case a = 1: the displayed bound p^{-1/2}(1+r(p) sqrt p) <= p^{-kappa} must be checked against the actual definition of r(p) on its whole support.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 2's lower bound stands or falls on Lemma 10, and within Lemma 10 the decisive step is the treatment of nonzero frequencies with |l dot a| > R. The proof compresses this into inequality (20): U(a,b) <= r(a)K(b)S_X, justified by saying that after the parametrization \"the sum over n1,k is a subsum of S_X\". This is the load-bearing step. U is used for reduced pairs a/b coming from m/(nh) with nh <= H, whereas S_X only sums products <= X. The ratio H/X = T^{1/2-eta}/T^{1/2-delta} = T^{1/3-21eta} tends to infinity, so the subsum claim is only valid if additional cancellation or a different parametrization is supplied. The problem is especially acute for negative frequencies: then l dot a < 0, so the reduced fraction has a/b = e^{2 pi l dot a} much less than 1 (up to a = 1 and b large), and the bound r(a)K(b) <= (ab)^{-kappa} from (21) is not derived; for a = 1 it would require K(b) <= b^{-kappa}, while K(b) = b^{-1/2} prod_{p|b}(1+r(p) sqrt p) contains factors that are not obviously <= p^{-kappa} on the stated support. If (20) or (21) fails in this regime, the o(S_X) bound for the high-frequency terms collapses, and the S_X/E_X lower bound that drives Theorem 2 is no longer available. The authors themselves describe the final phase of Theorem 2 as a \"brief comment\"; this is the specific estimate that the brevity conceals.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek/deepseek-v4-flash","summary":"The paper proves two unconditional statements about the Riemann zeta function on the zero set of a fixed real-rooted trigonometric polynomial P. Theorem 1 asserts that, among zeros of P in a dyadic interval [T,2T], the proportion that are also zeros of zeta on the critical line has limsup at most 2/3. Theorem 2 asserts that the maximum of |zeta(1/2+it)| on such zeros satisfies log sup >= (1+o(1)) sqrt(log T/(6 log log T)). The method replaces Poisson summation for arithmetic progressions by the Kurasov-Sarnak crystalline summation formula for the zero-counting measure of P. The first and second moments of zeta times a mollifier over the zeros of P are compared with their continuous analogues (Proposition 5), and Theorem 1 follows from the Balasubramanian-Conrey-Heath-Brown second moment bound. Theorem 2 uses a resonator with small exceptional primes removed, together with a spectral decomposition in which low crystalline frequencies are killed by the exceptional primes and high frequencies are controlled by coefficient decay of the crystalline measure.","tokens_in":24162,"tokens_out":1005,"duration_ms":645224,"significance":"The results are substantial if correct: they extend the arithmetic-progression theorems of Li and Radziwill to aperiodic zero sets, and they appear to be the first analytic number theory application of the Kurasov-Sarnak/Alon-Cohen-Vinzant crystalline measure machinery. The proof of Proposition 5 is a genuine discrete-to-continuous moment comparison and is of independent interest, as is the self-contained proof of Proposition 7. I specifically checked the delicate high-frequency estimate in Lemma 10: inequality (20) is valid, and the concern that H/X tends to infinity does not break the subsum argument, because the relevant constraint is m = a u n1 k <= X rather than the A_Q variable h <= TQ. The main remaining issues are expositional: a load-bearing step is compressed, and a few parameter and notational choices would benefit from being written out.","major_comments":[],"minor_comments":[{"comment":"Please expand the proof of (20). After the parametrization, for fixed u one has the summand r(a) r(u)^2 r(n1)^2 r(k) / sqrt(b1 k). Setting N = n1 and K = k identifies the (n1,k)-sum as a subsum of S_X, because a u n1 k <= X implies N K <= X/a; the u-sum is then bounded by b^{-1/2} prod_{p|b, r(p)ne 0} (1 + r(p) sqrt p). This argument also covers the negative-frequency case a = 1, b large, since the same local factor bound gives K(b) <= b^{-kappa}. The current sentence 'the sum over n1,k is a subsum of S_X' is correct but too terse for a load-bearing estimate.","section":"Section 6, Lemma 10, Eq. (20)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"There is a notational collision: a is both the vector in Z_M^+ indexing the crystalline frequencies and the numerator of the reduced fraction a/b arising from m/(nh). This makes (19)-(21) unnecessarily hard to read; please use distinct letters for the reduced numerator and denominator.","section":"Section 6, Step 1","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The dyadic decomposition should first dispose of the case M_a < K_a, in which the portion s >= K_a is empty. As written, N = ceil(log_2(M_a/K_a)) is not meaningful when M_a/K_a < 1, even though the subsequent estimate is valid after this case is excluded.","section":"Section 4, Step 3, R_{B,+}","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The final 'brief comment' is quite compressed. Please write out the two displayed steps: how (23) and (24) convert |M1|/M0 into a lower bound for sup |zeta|, and how X = T^{1/6 - 20 eta} leads to the constant 1/sqrt(6) after letting eta tend to 0. The argument is sound but currently relies on an implicit computation.","section":"Section 6, final paragraph","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"minor_revision","confidential_remarks":"I found no citation-integrity or novelty-disclosure issue. The central delicate estimate, inequality (20), checks out after spelling out the indexing; the revision needed is expositional rather than substantive."},"author_rebuttal":null,"desk_editor":{"model":"deepseek/deepseek-v4-flash","letter":"Short version: this is a real extension of Li-Radziwill to aperiodic zero sets, the Kurasov-Sarnak summation formula is doing substantive work, and the main theorems look plausible. I would send it out.\n\nWhat is new: the observation that arithmetic progressions are the periodic special case of zeros of sin(pi(x-b)/a), and the replacement of Poisson summation by the crystalline Fourier summation formula for the zero measure of a real-rooted trigonometric polynomial. The aperiodic case genuinely does not reduce to [13]. Theorem 1 is argued in detail: the good/bad frequency splits, the Farey-spacing uniqueness argument, and the R_{B,+}/R_{B,-} bounds are internally consistent in my spot checks, and the continuous second moment is an external benchmark from Balasubramanian-Conrey-Heath-Brown. The 2/3 constant emerges from theta to 1/2 without circularity.\n\nThe soft spots are concentrated in Section 6. The proof of Lemma 10 is the load-bearing part of Theorem 2, and it is compressed. I checked the specific worry about inequality (20) and negative frequencies. That objection does not stick. The parametrization in (20) is legitimate once one uses squarefree support: nonzero summands force gcd(n1,k)=1, so r(n1)^2 r(k) = r(n1) r(n1 k), exactly a subsum of S_X. And (21) is derived from the two local bounds r(p) <= p^{-kappa} and p^{-1/2}(1+r(p) sqrt p) <= p^{-kappa}, which hold uniformly on the stated support; the a=1, b large case is covered by the same product bound. So the high-frequency error bound is not as fragile as the stress-test note claims.\n\nThat said, the nh > H range is still too terse. The claim that the spacing condition leaves at most T^{eta+o(1)} integers h should be written out with the original h <= T^Q truncation made explicit, and the final exponent bookkeeping deserves a careful referee. The same referee should verify Lemma 9, since its one-line adaptation of [11, Prop 2.4] carries the Barban-Vehov input. Proposition 7 is reconstructed with an independent proof; the AI-assistance disclosure does not bother me as long as the argument is checked.\n\nFor whom: analytic number theorists working on zeta extremes, plus people interested in Fourier quasicrystals crossing into number theory. The paper deserves a serious referee, not a desk reject. I would take it to reading group and would cite it if I were working nearby.","headline":"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.","tokens_in":24618,"tokens_out":800,"would_cite":true,"duration_ms":72613,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":["11M06","11M26","42A05","42A38","34B45"],"pacs":[],"model":"deepseek/deepseek-v4-flash","headline":"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…","keywords":["Riemann zeta function","large values of zeta","real-rooted trigonometric polynomials","Fourier quasicrystals","crystalline measures","Lee-Yang polynomials","Kurasov-Sarnak summation formula","critical line"],"falsifier":"Pick a real-rooted trigonometric polynomial with aperiodic zero set — e.g., $P(x) = \\cos(2\\pi x) + \\cos(2\\pi\\sqrt{2}\\,x)$ — and, at growing $T$, compute the proportion (with multiplicity) of $P$'s zeros in $[T,2T]$ that are ordinates of zeros of $\\zeta$ on the critical line, using rigorous root-location of $\\zeta(1/2+it)$; any value above $2/3$ would contradict Theorem 1. A cheaper test targets Lemma 9: evaluate the Barban–Vehov weighted sums with the congruence condition $k\\equiv 1 \\pmod Q$ for small square-free $Q$ and moderate $X=Y$ to confirm the claimed main term $Q/\\phi(Q)\\log\\min(X,Y)$.","tokens_in":24047,"feed_emoji":"🧮","tokens_out":10187,"duration_ms":66044,"temperature":0.7,"pith_summary":"Fix any trigonometric polynomial P — a finite sum of exponentials with real frequencies — all of whose zeros are real. This paper proves two unconditional facts about the Riemann zeta function evaluated at those zeros on the critical line. First, at most two-thirds of P's zeros in any dyadic interval [T,2T] are ordinates of zeta zeros, in the limsup sense, counting multiplicity. Second, so long as P has infinitely many zeros, |ζ(1/2 + iγ)| exceeds exp((1+o(1))√(log T / (6 log log T))) at some zero γ of P in each such interval. The same two statements were previously known only when P's zero set is a union of arithmetic progressions; the novelty is that they now hold for every real-rooted such P, aperiodic zero sets included, and the proof uses crystalline Fourier summation in place of Poisson summation.","feed_headline":"Two-thirds zeta rule now holds for all real-rooted trig polynomials","feed_subtitle":"Unconditional non-coincidence and large-value bounds, aperiodic zero sets included.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the Kurasov–Sarnak Fourier summation formula (Theorem 4) and the coefficient bounds serving as the bridge between P's zeros and the analytic estimates.","marker":"[12]"},{"why":"Gives the Lee–Yang representation of every real-rooted trigonometric polynomial (representation (2)), the structural input to Theorem 4.","marker":"[1]"},{"why":"Provides the two arithmetic-progression theorems being extended, along with the mollification and resonator strategies adapted here.","marker":"[13]"},{"why":"Balasubramanian–Conrey–Heath-Brown second-moment bound for ζ·M_θ, used to close the Cauchy–Schwarz argument for Theorem 1.","marker":"[3]"},{"why":"Supplies the companion zero-count #zeros in [T,2T] = c₀T(1+o(1)) and the gap-distribution background.","marker":"[2]"},{"why":"Kumar Murty's Barban–Vehov theorem in arithmetic progressions, the basis of Lemma 9 via a one-line adaptation to the ρ_X weights and the condition k ≡ 1 mod Q.","marker":"[11]"},{"why":"Titchmarsh's truncation of ζ (Theorem 4.11) used for the approximate functional equation and the resonator-versus-zeta comparison.","marker":"[19]"},{"why":"Soundararajan's resonance estimate (Rankin's trick) used to lower-bound S_X/E_X in the proof of Theorem 2.","marker":"[17]"}],"fun_headline_variants":["At least one third of zeros of any real-rooted trig polynomial are zeta-free","Zeta avoids one third of every real-rooted trig polynomial zero set","Same zeta zero bounds now cover all real-rooted trig polynomials","Two-thirds coincidence bound extended beyond arithmetic progressions"],"cache_read_input_tokens":0,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["At least one third of zeros of any real-rooted trig polynomial are zeta-free","Zeta avoids one third of every real-rooted trig polynomial zero set","Same zeta zero bounds now cover all real-rooted trig polynomials","Two-thirds coincidence bound extended beyond arithmetic progressions"]},"model":"deepseek/deepseek-v4-flash","effort":"low","cost_usd":0.000191,"raw_usage":{"cost":0.0001911,"is_byok":false,"cost_details":{"upstream_inference_cost":0.0001911,"upstream_inference_prompt_cost":3.6946e-05,"upstream_inference_completions_cost":0.000154154},"total_tokens":1973,"prompt_tokens":1126,"completion_tokens":847,"prompt_tokens_details":{"audio_tokens":0,"video_tokens":0,"cached_tokens":900,"cache_write_tokens":0},"completion_tokens_details":{"audio_tokens":0,"image_tokens":0,"reasoning_tokens":781}},"tokens_in":1126,"tokens_out":847,"cost_source":"openrouter_usage","duration_ms":8402,"image_pages":[],"temperature":1.0,"billed_attempts":1,"reasoning_tokens":781,"transport_provider":"openrouter","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-10-20T06:59:01.544288+00:00","model_set":{"reader":"deepseek/deepseek-v4-flash"},"falsifier":"Pick a real-rooted trigonometric polynomial with aperiodic zero set — e.g., $P(x) = \\cos(2\\pi x) + \\cos(2\\pi\\sqrt{2}\\,x)$ — and, at growing $T$, compute the proportion (with multiplicity) of $P$'s zeros in $[T,2T]$ that are ordinates of zeros of $\\zeta$ on the critical line, using rigorous root-location of $\\zeta(1/2+it)$; any value above $2/3$ would contradict Theorem 1. A cheaper test targets Lemma 9: evaluate the Barban–Vehov weighted sums with the congruence condition $k\\equiv 1 \\pmod Q$ for small square-free $Q$ and moderate $X=Y$ to confirm the claimed main term $Q/\\phi(Q)\\log\\min(X,Y)$.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Kurasov–Sarnak Fourier summation formula (Theorem 4) and the coefficient bounds serving as the bridge between P's zeros and the analytic estimates."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the Lee–Yang representation of every real-rooted trigonometric polynomial (representation (2)), the structural input to Theorem 4."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the two arithmetic-progression theorems being extended, along with the mollification and resonator strategies adapted here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Balasubramanian–Conrey–Heath-Brown second-moment bound for ζ·M_θ, used to close the Cauchy–Schwarz argument for Theorem 1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the companion zero-count #zeros in [T,2T] = c₀T(1+o(1)) and the gap-distribution background."},{"cited_title":"Kumar Murty , The Barban--Vehov theorem in arithmetic progressions","cited_arxiv_id":null,"evidence_quote":"Kumar Murty's Barban–Vehov theorem in arithmetic progressions, the basis of Lemma 9 via a one-line adaptation to the ρ_X weights and the condition k ≡ 1 mod Q."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Titchmarsh's truncation of ζ (Theorem 4.11) used for the approximate functional equation and the resonator-versus-zeta comparison."},{"cited_title":"Soundararajan, Extreme values of zeta and L -functions, Math","cited_arxiv_id":null,"evidence_quote":"Soundararajan's resonance estimate (Rankin's trick) used to lower-bound S_X/E_X in the proof of Theorem 2."}],"review_version":1}