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arXiv · 2110.04278

On Large Values of $|ζ(σ+{\rm i}t)|$

Zikang Dong·Bin Wei
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Abstract

We investigate the extreme values of the Riemann zeta function $ζ(s)$. On the 1-line, we obtain a lower bound evaluation $$\max_{t\in[1,T]}|ζ(1+ıt)|\ge {\rm e}^γ(\log_2T+\log_3T+c),$$ with an effective constant $c$ which improves the result of Aistleitner, Mahatab and Munsch. In the half-critical strip $1/2<\re s<1$, we get an improved $c(σ)$ in the evaluation $$\max_{t\in[0,T]}\log|ζ(σ+ıt)|\ge c(σ)\frac{(\log T)^{1-σ}}{(\log_2T)^σ},$$ when $σ\searrow 1/2$, based on an improved lower bound of GCD sums. This improves the result of Bondarenko and Seip.

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Zikang Dong, Bin Wei. 2022-03-12. On Large Values of $|ζ(σ+{\rm i}t)|$. https://arxiv.org/abs/2110.04278

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