Correlations of the Riemann zeta function

2023/03/17 by Michael Curran, Curran, Michael J. · 2 citations
Mathematics · #Advanced Mathematical Identities #Advanced Mathematical Theories #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2303.10123

openalex publication_date 2023/03/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Assuming the Riemann hypothesis, we investigate 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. We shall prove that 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}. This improves upon the previous best known bounds due to Chandee and Ng, Shen, and Wong, particularly when the differences |α_j - α_k| are unbounded as T → ∞. The key insight is to combine work of Heap, Radziwiłł, and Soundararajan and work of the author with the work of Harper on the moments of the zeta function.

Cited by

• A survey of moment bounds for ζ(s): from Heath Brown's work to the present
• Lower bounds for high moments of zeta sums

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