Extreme values of derivatives of the Riemann zeta function

2021/08/04 by Yang, Daodao · 2 citations
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2108.02301

Abstract

It is proved that if T is sufficiently large, then uniformly for all positive integers ℓ \leqslant (log T) / (log₂ T), we have max_{T\leqslant t\leqslant 2T}|ζ^(ℓ)(1+it)| \geqslant e^γ⋅ ℓ^ℓ⋅ (ℓ+1) ^-(ℓ+1)⋅(log₂ T - log₃ T + O(1))^ℓ+1 , where γ is the Euler constant. We also establish lower bounds for maximum of |ζ^(ℓ)(σ+it)| when ℓ ∈ \mathbb N and σ∈ [1/2, 1) are fixed.

Cited by

• Large values of derivatives of the Riemann zeta function on vertical homogeneous progressions
• Extreme values of derivatives of the Dedekind zeta function of a cyclotomic field

Related

• On Large Values of |ζ(σ+\rm it)|
• A note on the large values of |ζ^(ℓ)(1+\rm it)|
• On derivatives of zeta and L-functions near the 1-line
• Extreme values of the Riemann zeta function on the 1-line
• A note on the maximum of the Riemann zeta function on the 1-line