A heuristic for discrete mean values of the derivative of the Riemann zeta function

2023/05/23 by C. P. Hughes, Greg S. Martin, Hughes, Christopher +3 · 1 citation
Mathematics · #Analytic Number Theory Research #Meromorphic and Entire Functions #Advanced Mathematical Identities

paper · pdf · doi:10.48550/arxiv.2305.14253

Abstract

Shanks conjectured that ζ' (ρ), where ρ ranges over non-trivial zeros of the Riemann zeta function, is real and positive in the mean. We present a history of this problem, including a generalisation to all higher-order derivatives ζ⁽ⁿ⁾(s), for which the sign of the mean alternatives between positive for odd n and negative for even n. Furthermore, we give a simple heuristic that provides the leading term (including its sign) of the asymptotic formula for the average value of ζ⁽ⁿ⁾(ρ).

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• Integer moments of the derivatives of the Riemann zeta function

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