Fourier optimization and pair correlation problems

Antonio Pedro Ramos; Mithun Kumar Das; Tolibjon Ismoilov

REVIEW 1 major objections 6 minor 21 references

Fourier optimization and pair correlation problems

T0 review · 1 major / 6 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read A Fourier-optimization framework bounds pair-correlation averages and refutes a 2018 conjecture in zeta-zero cases.

desk verdict Genuinely new axiomatic framework for form-factor averages, with a clean conditional refutation of Gonek-Ki Conjecture 1; the core transfer is solid and the paper deserves serious peer review. read the letter →

arxiv 2502.05106 v1 pith:DAJCLZDQ submitted 2025-02-07 math.NT math.CA

Mithun Kumar Das , Tolibjon Ismoilov , Antonio Pedro Ramos This is my paper · ORCID

classification math.NTmath.CA MSC 41A3046E2211M0611M26

keywords paircorrelationformfactorFourieroptimizationreproducingkernelHilbertspaceRiemannzetafunctionSelbergclassDedekindextremalproblems

topics The Riemann Hypothesis

open problems The Riemann Hypothesis

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to convert pair-correlation questions for arbitrary sequences of real numbers into two extremal problems in Fourier analysis. Under a density axiom (A1) and a weak-* convergence axiom (A2) for the associated form-factor measure, it shows that long averages of the form factor are sandwiched between $1+s_0(C_\nu-1)$ and $C_\nu$, where $s_0=-0.217\ldots$ is the minimum of $\sin x/x$ and $C_\nu$ is an optimization constant depending only on the limiting measure $\nu$. For the measures that occur for zeta-family zeros, $d\nu(\alpha)=c_1\delta(\alpha)+c_2|\alpha|e^{-c_3|\alpha|}d\alpha$, the paper computes an upper bound for $C_\nu$ through reproducing-kernel Hilbert spaces, yielding explicit numerical intervals for Selberg-class zeros, Dedekind zeta zeros, and zeros of the real and imaginary parts of the Riemann zeta function. In the last case the long averages stay above $0.7467$, which contradicts the 2018 conjecture that they should tend to zero for fixed $c>0$. A second, exactly solved extremal problem gives a universal lower bound of $1/2$ for symmetric averages, independent of the limiting measure.

What carries the argument

The load-bearing object is the extremal constant $C_\nu=\inf_{g\in A_\Delta,\,g(0)>0}\Phi_\nu(g)/g(0)$, where $A_\Delta$ consists of continuous even functions $g$ with $g,\hat g\in L^1$, $g\ge 0$, and $\hat g(\alpha)\le 0$ outside $[-\Delta,\Delta]$, and $\Phi_\nu(g)=\int_{-\Delta}^{\Delta}\hat g(\alpha)\,d\nu(\alpha)$. For measures $\nu=c_1\delta+c_2|\alpha|e^{-c_3|\alpha|}d\alpha$ supported in $[-\Delta,\Delta]$, the paper works in the reproducing kernel Hilbert space $H_\nu$ of entire functions of exponential type at most $\pi\Delta$ with inner product weighted by $\hat\nu$; a reproducing kernel Hilbert space is one in which each point evaluation is a continuous functional, represented by a kernel function $K_\nu$. Lemma 4 gives $C_\nu\le 1/K_\nu(0,0)$, and Theorems 5 and 6 compute $K_\nu(0,0)$ by solving the integral equation $(\hat f*\nu)=e^{-2\pi i w\xi}$ with ordinary differential equations. The auxiliary problem EP2, maximizing $g(0)$ subject to $\hat g\le \chi_{[-\beta,\beta]}$ and $g\ge 0$, is solved exactly by $D_\beta=\beta$ using Poisson summation, producing the universal $1/2$ lower bound.

What would settle it

Numerically compute, for fixed $c>0$ and large $b,\ell,T$, the averages $(1/\ell)\int_b^{b+\ell}F_\Gamma(\alpha,T)\,d\alpha$ for zeros of $\operatorname{Re}\zeta$ or $\operatorname{Im}\zeta$ on the line $a=1/2-c/\log T$, using rigorous high-precision zero data; if they converge to $0$ as the conjecture predicts, the lower constant $0.7467$ is false. A more direct check is to test (A2) itself by confirming whether $F_\Gamma(\alpha,T)\,d\alpha$ on $[-1/2,1/2]$ converges weak-* to $\delta(\alpha)+|\alpha|e^{-4c|\alpha|}d\alpha$.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 1: any sequence satisfying (A1) and (A2) has form-factor averages obeying $1+s_0(C_\nu-1)-\varepsilon+o(1)<(1/\ell)\int_b^{b+\ell}F_\Gamma(\alpha,T)\,d\alpha<C_\nu+\varepsilon+o(1)$ as $T\to\infty$, with $\ell$ large depending on $\varepsilon$ and, for the lower bound, on $b$. The constant $C_\nu$ is defined by the extremal problem over even nonnegative test functions whose Fourier transform is nonpositive outside $[-\Delta,\Delta]$. For the applied measures $\nu=c_1\delta+c_2|\alpha|e^{-c_3|\alpha|}d\alpha$ with $c_2\Delta^2/c_1\le 5/3$, the paper constructs the reproducing kernel Hilbert space $H_\nu=PW(\pi\Delta)$ and proves $C_\nu\le 1/K_\nu(0,0)$, then solves the kernel's differential equation to get explicit values. Feeding in previously established form-factor asymptotics for primitive Selberg-class functions, Dedekind zeta functions, and the real and imaginary parts of zeta, it obtains Corollaries 8 through 10. Corollary 10's lower bound $0.7467\ldots$ is uniformly positive on long intervals, so the conjecture's predicted vanishing of the averages for fixed $c>0$ cannot hold.

Load-bearing premise

The hinge is Assumption (A2): the form-factor measures $F_\Gamma(\alpha,T)\,d\alpha$ must converge weak-* to a finite limiting measure $\nu$; in the number-theoretic applications this convergence is imported from unproved input such as the Riemann Hypothesis, so if that input is wrong the bounds and the disproof of the conjecture do not follow.

Editorial extensions

If this is right

Any sequence satisfying (A1)–(A2) has long form-factor averages trapped between $1+s_0(C_\nu-1)$ and $C_\nu$, forcing $C_\nu\ge 1$ as a necessary condition on any limiting measure.
For symmetric intervals, the density axiom alone forces long averages to be at least $1/2$, regardless of the size of the interval or the shape of the limiting measure.
For primitive Selberg-class functions of degree $m$, the averages are explicitly bounded in terms of $m$; for Dedekind zeta functions of abelian degree-$n$ fields, in terms of $n$.
For zeros of the real and imaginary parts of zeta with $a=1/2-c/\log T$, the averages lie between $0.7467$ and $2.1659$, so the addressed conjecture cannot hold for fixed $c>0$; whether it can hold when $c=o(1)$ is left open.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: the same reproducing-kernel computation should apply, with the same explicit constants, to any sequence whose limiting measure has the form $c_1\delta+c_2|\alpha|e^{-c_3|\alpha|}d\alpha$, once an (A2)-type input is available.
Editorial inference: the universal $1/2$ lower bound suggests a rigidity of pair correlations that might hold under weaker hypotheses than (A2), perhaps whenever a positive proportion of the form factor's mass stays near $\alpha=0$.
Editorial inference: a numerical experiment comparing the Re/Im zeta averages with the conjecture's predicted value would locate the breakdown, either in the conjectured form factor or in the RH-supplied input, and so would tell future work where to focus.
Editorial inference: optimizing the kernel parameter $c_3$ and the interval $[-\Delta,\Delta]$ is a natural next step, since the bounds are valid for all admissible parameters and their tightness varies with them.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

I read the paper carefully. The main new thing is Theorem 1: from two axioms, one about the counting function and one about weak-* convergence of the form factor to a measure ν, you get explicit long-term average bounds for FΓ in terms of a Fourier optimization constant Cν. That's a genuine unification: Montgomery, Selberg-class, Dedekind zeta, and Gonek-Ki's Re/Im ζ cases all fall out of the same machine. The proof is a clean adaptation of the Carneiro-Milinovich-Ramos averaging mechanism, and I see no gap there.

The exact solution Dβ = β for the second extremal problem is a nice little result, and Theorem 2's unconditional 1/2 lower bound is a useful byproduct. The RKHS section is heavier. I was initially startled by the closed form in Theorem 5 because the text extraction shows a plus sign where a fraction bar belongs. Working from the boundary-value problem gives Kν(0,0) = √(2/(c1c2)) sin x / (cos x + √(c2/(2c1)) Δ sin x), with x = √(c2/(2c1)) Δ, and that exactly reproduces the constants in Corollaries 8 and 10 (2.1659 and 0.7467 for Δ=1/2, c1=c2=1). So the paper is internally consistent; someone with the PDF should just confirm the displayed formula isn't garbled in the original. The c3>0 formula in Theorem 6 is a serious ODE computation; I didn't verify every line, but the appendix's non-vanishing lemma covers the parameter range actually used, and the limiting cases are handled.

The refutation of Gonek-Ki Conjecture 1 is the headline. The limiting measure from their own Theorem 3 is δ + |α|e^{-4c|α|}, and Corollary 10 gives a lower bound that stays above 0.7467 for fixed c, while the conjectured expression averaged over long intervals [b,b+ℓ] tends to 0 as b or ℓ grows. The contradiction is real. It is conditional on RH because the input asymptotics are, but the conjecture lives in that same conditional world, so that's not a flaw.

Soft spots: the supremum sup_{t>0}(2-2cos t-2t sin t)/t² = 0.586... is quoted without derivation. It's elementary and not load-bearing, since all applications have c2/c1 Δ² well below 5/3. Theorem 6's proof delegates substantial algebra to ODE solving; an independent check would be welcome, but I found no internal contradiction and the numerics match.

I'd send this to a serious referee. It's a real contribution, not a routine extension. My own verdict would be conditional accept, pending a clean statement of Theorem 5 and a verification of the numerical constants in the c3>0 case.

Referee Report

1 major / 6 minor

Summary. The paper introduces an axiomatic framework, Assumptions A1 and A2, for the form factors F_Γ(α,T) of a general family of sequences Γ(T), and proves that long-interval averages of F_Γ are controlled by the Fourier-optimization constant C_ν attached to the limiting measure ν. Theorem 1 gives the bounds 1+s0(C_ν−1)−ε+o(1) < (1/ℓ)∫_b^{b+ℓ}F_Γ dα < C_ν+ε+o(1), and Theorem 2 gives a universal lower bound of 1/2 via the complete solution of a second extremal problem (EP2). For measures of the form dν=c1δ+c2|α|e^{-c3|α|}dα, the paper constructs the reproducing kernel Hilbert space H_ν, computes K_ν(0,0) (Theorems 5 and 6), and derives explicit average bounds for zeros of Selberg-class L-functions, Dedekind zeta functions, and the real and imaginary parts of the Riemann zeta function. In the last case, Corollary 10 yields a positive lower bound for averages of the form factor, which is incompatible with Gonek and Ki's Conjecture 1 when c>0 is fixed.

Significance. If correct, the paper provides a clean and broadly applicable transfer principle from Fourier optimization to pair-correlation averages, and the disproof of Gonek–Ki Conjecture 1 is a notable result. The solution of EP2 is complete and elementary, and the RKHS approach gives explicit, reproducible constants. The framework is transparently axiomatic: the burden is placed on external theorems that supply A2, and the paper is careful to state this dependence. The numerical lower bound 0.7467 for the averages attached to Re ζ and Im ζ is a concrete, falsifiable consequence, and the internal consistency of the numerical constants in Corollaries 8–10 is a strength.

major comments (1)

[§1.5, Theorem 5] The displayed formula for K_ν(0,0) is inconsistent with the rest of the paper as written. The formula appears to state K_ν(0,0)=√(2/(c1c2)) sin θ cos θ + θ sin θ, where θ=√(c2/(2c1))Δ. For Δ=1, c1=c2=1 this would give 1/K_ν(0,0)≈0.864, contradicting Theorem 2's lower bound of 1/2 and the known Montgomery–Carneiro–Milinovich–Ramos bounds for the zeta function. The numerical values in Corollaries 8–10 correspond instead to K_ν(0,0)=√(2/(c1c2)) sin θ / (cos θ + θ sin θ). Since Theorem 5 is the basis for the explicit constants in all applications, the displayed formula must be corrected and checked.

minor comments (6)

[§2.1] The class ABL_Δ is used in the proof of Theorem 1 before being defined; please state explicitly that ABL_Δ consists of those f∈A_Δ with supp(f̂) compact.
[§3.1] The numerical bound sup_{t>0}(2−2cos t−2t sin t)/t^2 = 0.586... is load-bearing for the admissibility range c2/c1 Δ^2 ≤ 5/3, but no proof or reference is supplied; include a short derivation or a citation.
[§4.1, Theorem 13] There is a typographical artifact in the definition of r(z): the formula ends with a stray semicolon, and the surrounding display would be clearer with consistent notation for the limit cases.
[§1.6.3] In the sentence introducing Gonek and Ki's Theorem 3, there is a stray '1.' after 'established the following'; this should be removed.
[§1.5, Theorem 6] The statement 'where the numbers η1 and η2 are roots of the equation ... such that η1+η2≠0' leaves the choice of roots ambiguous; the proof uses one root from each of the two pairs {±η1}, {±η2}, and this convention should be stated in the theorem.
[Corollaries 8 and 9] The displayed lower bounds are difficult to parse because of missing parentheses and line breaks (e.g., '− 2m − 1 2m' should presumably be '− (2m−1)/(2m)'). Please typeset these expressions with clear fraction bars and parentheses.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: bounds transfer from external limiting measures via Fourier optimization; the Gonek-Ki disproof is a valid modus tollens.

full rationale

The derivation chain is A1/A2 -> Theorem 1 -> EP1 constant C_nu -> Lemma 4 (C_nu <= 1/K_nu) -> Theorems 5 and 6 (K_nu) -> Corollaries 7-10 -> contradiction with Gonek-Ki Conjecture 1. No step defines a target quantity in terms of itself. Assumption A2 is an axiom in the abstract framework, but in every application it is supplied by external, independently proven theorems: Montgomery/Goldston-Montgomery for zeta, Murty-Perelli under GRH plus Hypothesis MP for the Selberg class, de Laat et al. under GRH for Dedekind zeta, and Gonek-Ki Theorem 3 under RH for Re/Im zeta. A2 is not constructed from the long-interval averages that the paper bounds. The constant C_nu is an extremal quantity over test functions g determined by the limiting measure nu alone; Theorem 1 transfers control of the integral of g-hat against F_Gamma to long intervals via Fourier inversion, non-negativity of g, the sign condition on g-hat, and the diagonal/off-diagonal splitting with s0. This is a genuine analytic transfer, not a substitution of the target average into its own definition. The reproducing kernel K_nu is obtained by solving the independent integral equation (u_nu * nu)(xi) = e^{-2 pi i w xi}; it depends only on nu and does not encode the interval averages being estimated. The disproof of Gonek-Ki Conjecture 1 is a legitimate modus tollens: assuming the conjecture, integrating its predicted expression over [b,b+ell] gives averages tending to 0, while Corollary 10's lower bound is derived from the proved small-alpha theorem of Gonek-Ki (via A2) together with the Fourier-optimization machinery. Thus the conjecture is contradicted by a theorem, not by itself. The reused averaging inequality from [5] (with co-author Ramos) is a published, checkable lemma and is independent support; it is not a fitted parameter and does not encode the target result. One non-circular completeness gap is that the numerical supremum 0.586... in Section 3.1 is asserted without derivation, but all applications use parameter values well inside the resulting 5/3 range, so this gap is not load-bearing. Verdict: no significant circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no fitted numerical parameters. The constants c1, c2, c3, and Delta are determined by the known limiting measures of the sequences, and the range condition c2/c1 Delta^2 <= 5/3 is a sufficient hypothesis for the positivity argument, not a fit. The main unproved inputs are the two axioms and the external conditional theorems that supply them.

assumptions (5)

domain assumption A1: #{gamma in Gamma(T) : 0 < gamma <= T} ~ lambda T log T / (2 pi)
Section 1.2, Assumption 1: the sequence has the standard density; needed to normalize FGamma and obtain the o(1) terms in the averaging arguments.
domain assumption A2: weak-* convergence of FGamma dalpha to a finite limiting measure nu on [-Delta, Delta]
Section 1.2, Assumption 2: imported from RH/GRH and external theorems in the applications; if the input asymptotics fail, the corresponding bounds do not follow.
domain assumption RH/GRH and Murty-Perelli Hypothesis MP in the Selberg-class and Dedekind applications
Section 1.6: Corollaries 8 and 9 are stated under GRH and Hypothesis MP; Corollary 10 is stated under RH.
ad hoc to paper Range condition c2/c1 Delta^2 <= 5/3 for the measure class studied
Theorem 3: a sufficient condition for the positivity of the Fourier transform of nu; the paper notes the method can slightly extend the range but chooses 5/3 for simplicity.
standard math Standard Fourier analysis, Paley-Wiener theorem, Krein decomposition, and Poisson summation
Used throughout Sections 2-4 to pass between test functions, form factors, and reproducing kernel Hilbert spaces.

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Cite this review

Pith. "Pith review of Fourier optimization and pair correlation problems." pith.science (2026). https://pith.science/paper/DAJCLZDQ

@misc{pith2026250205106,
  author       = {Pith},
  title        = {Pith review of: Fourier optimization and pair correlation problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DAJCLZDQ}},
  note         = {Machine review of arXiv:2502.05106}
}

read the original abstract

We introduce a generic framework to provide bounds related to the pair correlation of sequences belonging to a wide class. We consider analogues of Montgomery's form factor for zeros of the Riemann zeta function in the case of arbitrary sequences satisfying some basic assumptions, and connect their estimation to two extremal problems in Fourier analysis, which are promptly studied. As applications, we provide average bounds of form factors related to some sequences of number theoretic interest, such as the zeros of primitive elements of the Selberg class, Dedekind zeta functions, and the real and imaginary parts of the Riemann zeta function. In the last case, our results bear an implication to a conjecture of Gonek and Ki (2018), showing it cannot hold in some situations.

Figures

Figures reproduced from arXiv: 2502.05106 by the authors.

**Figure 1.** Upper and lower bounds for 1 ℓ R b+ℓ b FΓ(α, T ) dα as a function of c, where c1 = 1, c2 = 1, c3 = 4c, and ∆ = 0.5. For the form factor associated with the zeros of fa(w, θ), Gonek and Ki [15, Theorem 3] established the following 1 . Let 0 < δ < 1/2. Under RH, for T ≥ T0 and δ ≤ a ≤ 1/2, FΓ(α, T ) ∼ 2T −4|α| log T + 2|α| (a + 1/2)(3 − 2a) T (4a−2)|α| , as T → ∞, uniformly for 0 ≤ |α| ≤ 1 2 − δ and δ ≤ a ≤ 1/2. From … view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

[1] [1]

N. I. Achieser, Theory of Approximation, Frederick Unga r Publishing, New York, 1956

work page 1956

[2] [2]

Barrett, B

W. Barrett, B. McDonald, S. J. Miller, P. Ryan, C. L. Turna ge-Butterbaugh, K. Winsor, Gaps between zeros of GL(2) L-functions, J. Math. Anal. Appl. 429 (2015), no. 1, 204–232

work page 2015

[3] [3]

Carneiro, V

E. Carneiro, V. Chandee, A. Chirre, and M. B. Milinovich, On Montgomery’s pair correlation conjecture: a tale of three integrals, J. Reine Angew. Math. 786 (2022), 2 05–243

work page 2022

[4] [4]

Carneiro, V

E. Carneiro, V. Chandee, F. Littmann, and M. B. Milinovic h, Hilbert spaces and the pair correlation of zeros of the Riemann zeta-function, J. Reine Angew. Math. 72 5 (2017), 143–182

work page 2017

[5] [5]

Carneiro, M

E. Carneiro, M. B. Milinovich, A. P. Ramos, Fourier optim ization and Montgomery’s pair correlation conjecture, Math. Comp. 94 (2025), 409–424

work page 2025

[6] [6]

Chandee, K

V. Chandee, K. Klinger-Logan, and X. Li, Pair correlatio n of zeros of Γ 1(q) L-functions, Math. Z. 302 (2022), no. 1, 219–258

work page 2022

[7] [7]

H. Cohn, N. Elkies, New upper bounds on sphere packings I, Ann. of Math. (2) 157 (2003), no. 2, 689–714

work page 2003

[8] [8]

Pair correlation for Dedekind zeta functions of abelian extensions

D. de Laat, L. Rolen, Z. Tripp, I. Wagner, Pair correlatio n for Dedekind zeta functions of abelian extensions, preprint available at https://arxiv.org/abs /1908.04876

work page Pith review arXiv 1908