Integer moments of the derivatives of the Riemann zeta function

Christopher Hughes; Andrew Pearce-Crump

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arXiv · 2509.07792

Integer moments of the derivatives of the Riemann zeta function

Christopher Hughes·Andrew Pearce-Crump

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Abstract

We conjecture the full asymptotic expansion of a product of Riemann zeta functions, evaluated at the non-trivial zeros of the zeta function, with shifts added in each argument. By taking derivatives with respect to these shifts, we form a conjecture for the integer moments of mixed derivatives of the zeta function. This generalises a result of the authors where they took complex moments of the first derivative of the zeta function, evaluated at the non-trivial zeros. We approach this problem in two different ways: the first uses a random matrix theory approach, and the second by the Ratios Conjecture of Conrey, Farmer, and Zirnbauer.

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Christopher Hughes, Andrew Pearce-Crump. 2025-09-09. Integer moments of the derivatives of the Riemann zeta function. https://arxiv.org/abs/2509.07792

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