Microscopic shift asymptotics for zeta-derivative moments matching the CUE Kostka–determinant formula

Establish that for lists of non‑negative integers μ and ν of equal length s and fixed c ∈ ℂ, the mean value (1/T) ∫_{1}^{T} ∏_{i=1}^{s} ζ^{(μ_i)}(σ_T + i t) ∏_{j=1}^{s} overline{ζ^{(ν_j)}(σ_T + i t)} dt with σ_T = 1/2 + c / log T has leading-order asymptotics given by A_{μ,ν} (log T)^{|μ| + |ν| + s^2} times the universal coefficient μ! ν! ∑_{λ ⊢ |μ|, ℓ(λ) ≤ s} ∑_{ρ ⊢ |ν|, ℓ(ρ) ≤ s} K_{λ μ} K_{ρ ν} λ!_{(s)} ρ!_{(s)} det(I_{α_i(λ)+β_j(ρ)}(τ))_{i,j=1}^{s}, where α_i(λ)=λ_i + s − i, β_j(ρ)=ρ_j + s − j, τ = c + ar c, I_r(τ)=∫_{0}^{1} x^{r} e^{−τ x} dx, and A_{μ,ν} = ∏_{p} {(1 − p^{−1})^{s^2} ∑_{m=0}^{∞} (Γ(s+m)/(m! Γ(s)))^2 p^{−m}}.

Background

The paper proves large-N asymptotics for joint moments of higher derivatives of CUE characteristic polynomials. In the microscopic regime z = 1 − c/N with |z| near the unit circle, Theorem 1.2 (Theorem \ref{thm:micro2}) shows that M_{μ,ν}(z,N) ~ N^{|μ|+|ν|+s^2} times a universal coefficient expressed as a sum over partitions with Kostka numbers and a determinant of integrals I_r(τ).

On the number-theoretic side, the authors analyze mean values of products of derivatives of the Riemann zeta function off the critical line. Assuming the Lindelöf hypothesis (and unconditionally for some low-length cases), they show that as σ→1/2^+ the leading term agrees with the CUE prediction in a mesoscopic regime, multiplied by an arithmetic Euler product factor.

They explicitly conjecture that in the microscopic scaling σ_T = 1/2 + c/log T, the leading coefficients of these zeta mean values match the microscopic CUE expression, with N replaced by log T and the same arithmetic Euler product factor. This would extend the CUE–zeta correspondence for higher derivative moments to the microscopic scale.