Lower bounds for shifted moments of the Riemann zeta function

2024/05/14 by Michael Curran, Curran, Michael J.
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.2405.08725

openalex publication_date 2024/05/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In previous work, the author gave upper bounds for the shifted moments of the zeta function M_α,β(T) = ∫_T^2T ∏_{k = 1}^m |ζ(\tfrac12 + i (t + α_k))|^{2 β_k} dt introduced by Chandee, where α = α(T) = (α₁, …, α_m) and β = (β₁ … , β_m) satisfy |α_k| ≤ T/2 and β_k≥ 0. Assuming the Riemann hypothesis, we shall prove the corresponding lower bounds: M_α,β(T) ≫_β T (log T)^{β₁2 + ⋯ + β_m2} ∏_{1≤ j lt; k ≤ m} |ζ(1 + i(α_j - α_k) + 1/ log T )|^{2β_j β_k}.

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