Higher order derivative moments of CUE characteristic polynomials and the Riemann zeta function

Abstract: We use random matrix theory for the Circular Unitary Ensemble (CUE) to study moments of derivatives of the Riemann zeta function shifted a small distance from the critical line. The corresponding CUE moments are studied in the limit of large matrix size in two regimes: when the spectral parameter is (1) suitably far inside the unit disc, and (2) at a small distance from the unit circle. In case (1), we obtain an asymptotic formula as a combinatorial sum over contingency tables, while in case (2) we obtain a sum over certain determinants with multiplicative coefficients given by Kostka numbers. The latter result is also valid exactly on the unit circle. Then, we consider the analogous problem for mean values of derivatives of the zeta function with suitable shifts. Assuming the Lindelöf hypothesis, we show that this mean value gives rise to the same sum over contingency tables obtained in the CUE. For sufficiently low-order moments, we establish this result unconditionally.

• The paper establishes precise asymptotic formulas for joint higher-order derivative moments of CUE characteristic polynomials, uncovering intricate combinatorial structures.
• It leverages random matrix theory and analytic number theory to connect contingency table sums with Kostka-number weighted determinantal expressions.
• It extends the CUE–zeta function correspondence to shifted derivative moments, differentiating universal and arithmetic factors under hypotheses like Lindelöf.

Higher Order Derivative Moments of CUE Characteristic Polynomials and the Riemann Zeta Function

Introduction

This paper addresses the study of joint higher-order moments of derivatives of characteristic polynomials associated with the Circular Unitary Ensemble (CUE), and their analogues for shifted moments of the Riemann zeta function, both on and off the critical line. Leveraging modern random matrix theory as well as analytic number theory, the authors extend the established CUE–zeta function dictionary to encompass higher order derivatives, elucidating the precise combinatorial and asymptotic structures that arise.

Main Results in the CUE Regime

The primary object of study is the joint moment

$$M_{\mu,\nu}(z,N) := \mathbb{E}\left[\prod_{i=1}^{\ell(\mu)}\Lambda_N^{(\mu_i)}(z)\prod_{j=1}^{\ell(\nu)}\overline{\Lambda_N^{(\nu_j)}(z)}\right]$$

where the expectation is over Haar measure on $\mathrm{U}(N)$ and $\Lambda_N(z)$ is the characteristic polynomial.

Two asymptotic regimes are distinguished:

1. Mesoscopic Regime ($|z|<1-N^{-\alpha}$): For spectral parameters strictly inside the unit disc, an exact asymptotic formula is established. The moments are expressed as a weighted sum over pairs of non-negative integer matrices (contingency tables) with prescribed row and column sums, with explicit combinatorial coefficients $p_{n,m}(z,\bar{z})$ (see equation (2.7) in the paper). These asymptotics demonstrate the emergence of shifted magic squares in the characterization of mixed derivative moments.
2. Microscopic/Unit Circle Regime ($z\to 1$): Near the unit circle, including $|z| = 1$, the asymptotics transition to a new structure: a determinantal sum over Young tableaux shapes, weighted by products of Kostka numbers, with entries given by one-dimensional exponential integrals $I_r(\tau)$. This regime generalizes and unifies known results for low-order derivatives and connects to combinatorics of symmetric functions and representation theory. Strict positivity of these sums is established, which is nontrivial and ensures that leading power counts in $N$ are indeed sharp.

Notably, for length-2 lists $\mu=\nu=(k, k)$, closed-form evaluations are produced, yielding explicit combinatorial coefficients (Theorem/Lemma 2.8 and related corollaries). These formulas refine previous multiple-sum expressions for higher derivative moments on the unit circle.

Combinatorial and Representation-Theoretic Structure

A fundamental conceptual advance is the connection made between asymptotics for higher derivative moments and symmetric function theory, specifically through Schur function expansions and Kostka numbers. The authors rigorously derive that the derivative moments at microscopic scale (with several colliding derivatives) admit an explicit expansion over pairs of partitions $\mathrm{U}(N)$0 of the corresponding sum-of-orders, with the aforementioned weights (Theorem 2.11).

This structural insight clarifies the combinatorics underlying previous integral or sum formulas and, importantly, links these objects with representation-theoretic invariants such as the number of semistandard Young tableaux and magic squares. This point is further reinforced by the independent appearance of similar structures in concurrent works (e.g., [AAKP], as referenced).

Riemann Zeta Function Analogues

Building on the random matrix results, the paper systematically derives analogues for shifted high moments of the Riemann zeta function and its derivatives. For fixed $\mathrm{U}(N)$1, the corresponding problem is the mean value over $\mathrm{U}(N)$2 of shifted derivatives $\mathrm{U}(N)$3, with $\mathrm{U}(N)$4. Assuming the Lindelöf Hypothesis, the authors prove that the limiting zeta moments decompose into an "arithmetic factor" (an Euler product over primes) and a "universal factor" that exactly matches the random matrix contingency table sum obtained in the mesoscopic regime.

For moments of sufficiently low order ($\mathrm{U}(N)$5), these results are unconditional, relying only on classical mean-value theorems for Dirichlet polynomials and the boundedness of the error terms established by Titchmarsh and later refinements.

Explicit closed forms for the leading order coefficients for the case $\mathrm{U}(N)$6 are given. The calculation of these coefficients involves manipulations of multi-parameter combinatorial sums, yielding explicit expressions in terms of multinomial coefficients and binomial identities (Theorem 3.6).

Technical and Theoretical Implications

Numerical and Theoretical Contributions

• The asymptotic formulas are uniform in key parameters and cover previously intractable or only partially understood cases.
• The paper clarifies the precise transition between various scaling regimes in CUE moments, describing the crossover from contingency-table formulas well inside the unit disk to Kostka-number-weighted determinantal structures near and on the unit circle.
• For the Riemann zeta function, the results provide rigorous justification for previously conjectured forms of the "universal part" of shifted derivative moments (as $\mathrm{U}(N)$7 and $\mathrm{U}(N)$8), demarcating the random matrix universality and arithmetic independence.

Combinatorics and Symmetric Function Theory

• The explicit identification of Kostka numbers as the weight factors in the microscopic regime links random matrix theory with the combinatorial theory of symmetric functions, allowing for systematic derivation of explicit moment formulas.
• The strictly positive nature of these sums for all admissible $\mathrm{U}(N)$9 and $\Lambda_N(z)$0 is proved, filling a gap in prior literature concerning the sign and sharpness of leading-order asymptotics.

Number Theory and Lindelöf Hypothesis

• The results reinforce the predictive power of random matrix models for moments of the Riemann zeta function, not only for the function values themselves but for all joint orders of derivatives.
• The unconditional evaluation for moments of low length, and the conditional results for higher moments (under Lindelöf), give clear targets for further rigorous verification in analytic number theory.

Potential Directions for Future Research

• Extension of these methods and formulas to other symmetry types beyond CUE, such as GOE and GSE, and more general L-functions.
• Analysis of non-integer and fractional moments for higher derivatives, drawing connections with recent results for non-integer moments of $\Lambda_N(z)$1 and characteristic polynomials.
• Investigation of secondary terms beyond leading order, seeking connections with arithmetic statistics and potential connections with moments of $\Lambda_N(z)$2-functions in function fields.
• Deeper exploration of combinatorial identities and representation theory in the context of microscopic asymptotics for other random matrix or zeta-like models.

Conclusion

This work systematizes and extends the link between moments of high-order derivatives for CUE characteristic polynomials and the Riemann zeta function, establishing both new asymptotic formulas and new connections with combinatorial representation theory. The results offer a comprehensive framework for analyzing derivative moments in random matrix theory and analytic number theory, providing explicit expressions and structural insights that will inform future advances in both disciplines.