On derivatives of zeta and L-functions near the 1-line

2023/12/19 by Zikang Dong, Dong, Zikang, Yutong Song +5
Mathematics · #Analytic Number Theory Research #Analytic and geometric function theory #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2312.12199

openalex publication_date 2023/12/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the conditional upper bounds and extreme values of derivatives of the Riemann zeta function and Dirichlet L-functions near the 1-line. Let ℓ be a fixed natural number. We show that, if |σ-1|≪1/log₂t, then |ζ^(ℓ)(σ+ it)| has the same maximal order (up to the leading coefficients) as |ζ^(ℓ)(1+ it)| when t→∞. The range 1-σ≪1/log₂t is wide enough, since we also show that (1-σ) log₂t → ∞ (t → ∞) implies \limsup_t→∞|ζ^(ℓ)(σ+ it)| / (log₂t)^ℓ+1 = ∞. Similar results can be obtained for Dirichlet L-functions L^(ℓ)(σ,χ) with χ\pmod q.

Related

• Explicit estimates for the Riemann zeta function close to the 1-line
• Bounding zeta on the 1-line under the partial Riemann hypothesis
• Explicit bounds for the Riemann zeta-function on the 1-line
• Omega Theorems for Logarithmic Derivatives of Zeta and L-functions Near the 1-line
• A note on the large values of |ζ^(ℓ)(1+\rm it)|