Sharp bounds for joint moments of the Riemann zeta function

2024/03/01 by Michael Curran, Curran, Michael J., André Heycock +1
Mathematics · #Analytic Number Theory Research #Analytic and geometric function theory #FOS: Mathematics #Mathematical Inequalities and Applications #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2403.00902

openalex publication_date 2024/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In previous work, the first author obtained conjecturally sharp upper bounds for the joint moments of the (2k-2h)^th power of the Riemann zeta function with the 2h^th power of its derivative on the critical line in the range 1≤ k ≤ 2, 0 ≤ h ≤ 1. Unconditionally, we extend these upper bounds to all 0 ≤ h≤ k ≤ 2, and obtain lower bounds for all 0≤ h ≤ k+1/2. Assuming the Riemann hypothesis, we give sharp bounds for all 0≤ h ≤ k. We also prove upper bounds of the conjectured order for more general joint moments of zeta with its higher derivatives.

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