[论文解读] The discrete second moment of mixed derivatives of the Riemann zeta function

Benjamin Durkan, C. P. Hughes+1 more|Research Explorer (The University of Manchester)|Jan 9, 2026

Analytic Number Theory Research被引用 0

一句话总结

该论文给出在黎曼 zeta 函数的零处对离散二阶矩的全渐近展开，包含无条件误差项与RH 条件误差项。

ABSTRACT

We establish the full asymptotic for the discrete second moment of the Riemann zeta function of mixed derivatives evaluated at the zeta zeros, providing both unconditional and conditional error terms. This was first studied by Gonek, where only the leading order asymptotic was given, later extended by Conrey--Snaith and Milinovich to include the lower order terms for the first derivative. We extend the case of the first derivative to all derivatives.

研究动机与目标

• Motivate the study of the discrete moment I(mu,nu) = sum_{0<gamma<=T} zeta^{(mu)}(rho) zeta^{(nu)}(1-rho) and its relevance to zeta derivative moments.
• Extend prior leading-term results to a full asymptotic expansion with lower-order terms for all derivatives mu, nu.
• Generalize previous work (Gonek, Conrey–Snaith, Milinovich) to arbitrary derivatives and establish both unconditional and RH conditional results.
• Provide explicit coefficient structure for the asymptotic polynomial and relate it to Laurent coefficients of associated Dirichlet series.

提出的方法

• Apply Cauchy’s theorem on a rectangular contour to express I(mu,nu) as a contour integral.
• Use the functional equation to relate zeta^{(nu)}(1-s) to derivatives at s and obtain a convergent Dirichlet series expansion.
• Evaluate main terms via stationary phase and Perron’s formula to extract a polynomial in log(T/2π) of degree mu+nu+2.
• Compute coefficients C1^{(mu,nu)}(m,k) and C2^{(mu,nu)}(m,k) from Laurent expansions around s=1 and relate them to the final polynomial.
• Show the right-hand and left-hand vertical segments yield the same type of polynomial contributions, yielding the full asymptotic plus explicit error bounds.
• Under RH, obtain error term O(T^{1/2+ε}); unconditional error is O(T e^{-C√(log T)}).

(a) $\sum_{0<\gamma<T}\left|\zeta^{\prime}\left(\frac{1}{2}+i\gamma\right)\right|^{2}$

实验结果

研究问题

• RQ1 What is the full asymptotic expansion for the discrete second moment sum_{0<gamma<=T} zeta^{(mu)}(rho) zeta^{(nu)}(1-rho)?
• RQ2 How can one express the coefficients of the asymptotic polynomial in terms of Laurent coefficients c^{(mu,k)}_j and d^{(nu,k)}_j from the relevant Dirichlet series?
• RQ3 Do unconditional and RH-conditional error terms differ, and what are the precise forms of these errors?
• RQ4 Does the leading term agree with prior leading-order results (e.g., Gonek, Conrey–Snaith) for general mu, nu?

主要发现

• Theorem 1 gives sum_{0<gamma<=T} zeta^{(mu)}(rho) zeta^{(nu)}(1-rho) = (T/2π) P_{mu,nu}(log(T/2π)) + O(T e^{-C√log T}) unconditionally.
• The polynomial P_{mu,nu}(x) has degree mu+nu+2 with coefficients expressed through C1^{(mu,nu)}(m,k) and C2^{(mu,nu)}(m,k) built from Laurent coefficients c^{(mu,k)}_j and d^{(nu,k)}_j.
• Under the Riemann Hypothesis the error improves to O(T^{1/2+ε}) for any ε>0.
• Corollaries include an RH-confirmed discrete second moment expansion for all derivatives (Corollary 1) and recovery of Milinovich’s full asymptotic for mu=nu=1 (Corollary 2).
• The leading-order coefficient agrees with the previously known leading term in (1.2).
• Appendix examples illustrate explicit polynomials for specific derivative orders (e.g., second derivative).

(b) $\sum_{0<\gamma<T}\left|\zeta^{\prime}\left(\frac{1}{2}+i\gamma\right)\right|^{2}-\frac{1}{24\pi}T\left(\log\frac{T}{2\pi}\right)^{4}$