A note on "Higher order linear differential equations for unitary matrix integrals: applications and generalisations"

Abstract: In this note, we briefly introduce the background and motivation of the collaborative work [arXiv:2508.20797], and provide an outline of the main results. The latter relates to matrix and higher order scalar differential equations satisfied by certain Hankel and Toeplitz determinants involving I-Bessel functions, or equivalently certain unitary matrix integrals, and moreover puts this property in a broader context. We also investigate large gaps between zeros of the derivatives of the Hardy $\mathsf{Z}$-function, assuming the validity of a certain joint moments conjecture in random matrix theory.

• The paper derives higher-order linear differential equations to systematically generate required data for unitary matrix integrals, breaking a barrier posed by Painlevé equations.
• The study examines joint moments of derivatives of Hardy's Z-function and successfully obtains new bounds: larger than previously known, for large gaps between zeros of the function.
• Results provide quantitative measures detailing $\Theta^{(1)}$, $\Theta^{(2)} > 1.98$ and $1.71$ respectively, under reasonable moment conjectures.

Overview

This note by Forrester and Wei accompanies the collaborative work "Higher order linear differential equations for unitary matrix integrals: applications and generalisations" (Forrester et al., 28 Aug 2025). It serves two purposes: it situates the study of higher order linear differential equations satisfied by Hankel and Toeplitz determinants of I-Bessel functions — equivalently, certain unitary matrix integrals — within a broader mathematical context, and it presents two new conditional results on large gaps between zeros of derivatives of Hardy's $\mathsf{Z}$-function. The note is expository in its first part and computational-analytic in its second.

Random matrix theory and the moments of the zeta function

The number-theoretic motivation traces to Montgomery's pair correlation conjecture [M73], which coincides with Dyson's pair correlation for scaled CUE/GUE eigenvalues, and to the moment problem for $\zeta(1/2+it)$. The classical program seeks constants $g_l$ such that

$$\frac{1}{T}\int_0^T \left|\zeta\left(\tfrac{1}{2}+it\right)\right|^{2l} dt \sim c_l \frac{g_l}{\Gamma(1+l^2)} (\log T)^{l^2},$$

with $c_l$ an explicit Euler product. Known values are $g_1=1$ (Hardy–Littlewood), $g_2=2$ (Ingham), $g_3=42$ (Conrey–Ghosh conjecture), and $g_4=24024$ (Conrey–Gonek conjecture); no reliable conjecture exists for $l>4$ even under RH. Keating and Snaith's CUE characteristic polynomial computation [KS00] supplies the prediction $\zeta(1/2+it)$0, consistent with all known cases and supported by the hybrid Euler–Hadamard product of Gonek–Hughes–Keating [GHK07].

The generalization central to this work concerns joint moments of derivatives: for integers $\zeta(1/2+it)$1, $\zeta(1/2+it)$2, one compares

$\zeta(1/2+it)$3

(and its analogue with $\zeta(1/2+it)$4, the CUE counterpart of Hardy's real-valued $\zeta(1/2+it)$5-function) against corresponding zeta-function moments. Hughes established $\zeta(1/2+it)$6; Keating and Wei [KW24a] proved the general exponent $\zeta(1/2+it)$7 and expressed both leading coefficients via partition sums. Crucially, these coefficients reduce to derivatives at zero of the Hankel determinant $\zeta(1/2+it)$8, which itself rewrites as the unitary matrix integral

$\zeta(1/2+it)$9

A technical obstruction motivates the companion paper: as observed in [KW24b], beyond the power-series order needed for $g_l$0, the associated $g_l$1-Painlevé III$g_l$2 equation does not have a unique solution, so the Painlevé route cannot systematically generate the required data. The alternative structure — higher order linear differential equations — does not suffer from this non-uniqueness.

Linear differential equations from Selberg-type integrals

The eigenvalue form of the integral belongs to the Selberg correlation integral class, where the framework of Davis [Da68] and Forrester–Rains [FR12] yields an $g_l$3 matrix differential equation, hence a scalar linear ODE of degree $g_l$4. This applies equally to circular $g_l$5-ensemble generalizations with free determinant power. Two applications follow:

Combinatorics: since $g_l$6, where $g_l$7 counts permutations of $g_l$8 with longest increasing subsequence of length at most $g_l$9, the method recovers the degree-$$\frac{1}{T}\int_0^T \left|\zeta\left(\tfrac{1}{2}+it\right)\right|^{2l} dt \sim c_l \frac{g_l}{\Gamma(1+l^2)} (\log T)^{l^2},$$0 ODEs computed by Bergeron–Gascon [BG00] for $$\frac{1}{T}\int_0^T \left|\zeta\left(\tfrac{1}{2}+it\right)\right|^{2l} dt \sim c_l \frac{g_l}{\Gamma(1+l^2)} (\log T)^{l^2},$$1, extends them to arbitrary $$\frac{1}{T}\int_0^T \left|\zeta\left(\tfrac{1}{2}+it\right)\right|^{2l} dt \sim c_l \frac{g_l}{\Gamma(1+l^2)} (\log T)^{l^2},$$2, and yields a recursive algorithm for the coefficients $$\frac{1}{T}\int_0^T \left|\zeta\left(\tfrac{1}{2}+it\right)\right|^{2l} dt \sim c_l \frac{g_l}{\Gamma(1+l^2)} (\log T)^{l^2},$$3.

Number theory: the same formulation gives recursive relations for Taylor coefficients of the I-Bessel Hankel determinant at $$\frac{1}{T}\int_0^T \left|\zeta\left(\tfrac{1}{2}+it\right)\right|^{2l} dt \sim c_l \frac{g_l}{\Gamma(1+l^2)} (\log T)^{l^2},$$4 for any $$\frac{1}{T}\int_0^T \left|\zeta\left(\tfrac{1}{2}+it\right)\right|^{2l} dt \sim c_l \frac{g_l}{\Gamma(1+l^2)} (\log T)^{l^2},$$5, enabling efficient computation of $$\frac{1}{T}\int_0^T \left|\zeta\left(\tfrac{1}{2}+it\right)\right|^{2l} dt \sim c_l \frac{g_l}{\Gamma(1+l^2)} (\log T)^{l^2},$$6 and $$\frac{1}{T}\int_0^T \left|\zeta\left(\tfrac{1}{2}+it\right)\right|^{2l} dt \sim c_l \frac{g_l}{\Gamma(1+l^2)} (\log T)^{l^2},$$7 for large $$\frac{1}{T}\int_0^T \left|\zeta\left(\tfrac{1}{2}+it\right)\right|^{2l} dt \sim c_l \frac{g_l}{\Gamma(1+l^2)} (\log T)^{l^2},$$8 and fixed $$\frac{1}{T}\int_0^T \left|\zeta\left(\tfrac{1}{2}+it\right)\right|^{2l} dt \sim c_l \frac{g_l}{\Gamma(1+l^2)} (\log T)^{l^2},$$9 — precisely the data needed below.

Large gaps between zeros of $c_l$0

Define $c_l$1 over zeros of $c_l$2. Under RH, $c_l$3, introduced by Selberg, who proved $c_l$4; Conrey–Ghosh–Gonek obtained $c_l$5 (RH) and $c_l$6 (GRH), while Hall, assuming the joint moments conjecture at $c_l$7, $c_l$8, proved $c_l$9 average gaps. The widely conjectured value is $g_1=1$0, open even under GRH.

The paper's new results extend Hall's method to derivatives, using his generalized Wirtinger inequality together with an optimization over parameters $g_1=1$1: the maximal positive root $g_1=1$2 of a polynomial equation involving ratios $g_1=1$3 satisfies $g_1=1$4. Computing $g_1=1$5 and $g_1=1$6 for $g_1=1$7 via [KW24a, Theorem 24], the authors obtain:

Result	Assumption	Bound
$g_1=1$8	joint moments conjecture, $g_1=1$9, $g_2=2$0	improves CG ($g_2=2$1, RH), Hall ($g_2=2$2), Bui–Hall ($g_2=2$3)
$g_2=2$4	joint moments conjecture, $g_2=2$5, $g_2=2$6	improves Bui–Hall ($g_2=2$7)

The proofs verify positivity of the optimized parameters $g_2=2$8 (e.g., $g_2=2$9 for $g_3=42$0), ensuring strict monotonicity of $g_3=42$1 on $g_3=42$2 as required by Hall's inequality, yielding $g_3=42$3 and $g_3=42$4 respectively. These bounds are strictly sharper than all previously known ones, though they rest on unproved moment conjectures rather than on RH or GRH alone.

Limitations and open questions

The lower bounds in the two theorems are conditional: they assume the leading-order asymptotics of Conjecture 2 for specific derivative pairs, and no direct connection between that conjecture and the Riemann hypothesis is known. The computations use $g_3=42$5 only; larger $g_3=42$6 would improve the bounds, but requires $g_3=42$7 for $g_3=42$8, obtainable either from [KW24a] or more efficiently via the recursive Taylor-coefficient formulas of [KW24b] and [FW25]. The authors explicitly decline to pursue this extension, noting it follows from the same method. Whether $g_3=42$9 for any $g_4=24024$0, and whether the moment conjectures underlying these bounds can be established unconditionally, remain open.

Conclusion

The note clarifies why higher order linear differential equations, rather than Painlevé transcendents, are the appropriate integrable structure for computing derivative-moment coefficients of unitary matrix integrals, and demonstrates the practical payoff: explicit rational data for $g_4=24024$1 translate directly into record conditional lower bounds $g_4=24024$2 and $g_4=24024$3 for gaps between zeros of derivatives of Hardy's $g_4=24024$4-function.