论文标题
Zeta和$ L $ functions的衍生物的极端值
Extreme values of derivatives of zeta and $L$-functions
论文作者
论文摘要
事实证明，作为$ t \ to \ infty $，对于所有积极整数而言，均匀地表示$ \ ell \ leqslant（\ log_3 t） /（\ log_4 t）$，我们有\ begin \ begin {equation*} \ max_ 2t} \ left |ζ^{（\ ell）} \ big（1+it \ big）\ right | \ geqslant \ big（\ Mathbf y _ {\ ell}+ o \ left（1 \右） u^{\ ell}ρ（u）du $。这里$ρ（u）$是Dickman函数。我们有$ \ mathbf y _ {\ ell}> e^γ/（\ ell + 1）$和$ \ log \，\ Mathbf y _ _ {\ ell} = \ left（1 + o \ left（1 + o \ left（1 \ weles） Dirichlet $ L $ functions也建立了类似的结果。另一方面，在假设Riemann假设和广义的Riemann假设时，我们为$ \ left |建立了上限| ζ^{（\ ell）} \ left（1+it \ right）\ right | $和$ \ left | l^{（\ ell）}（1，χ）\ right | $。此外，当假设Granville-Soundararajan猜想是正确的时，我们建立以下渐近公式$$ \ max_ {\ setack {χ\ neq \ neq \ neqχ_0\\ \χ（\ text {mod} \，Q）}}}}}}}}}}}}}}}} _ \sim \mathbf Y_{\ell}\left(\log_2 q\right)^{\ell+1},\,\, \quad \text{as}\,\quad q \to \infty,$$ where $q$ is prime and $\ell \in \mathbb{N}$ is given.
It is proved that as $T \to \infty$, uniformly for all positive integers $\ell \leqslant (\log_3 T) / (\log_4 T)$, we have \begin{equation*} \max_{T\leqslant t\leqslant 2T}\left|ζ^{(\ell)}\Big(1+it\Big)\right| \geqslant \big(\mathbf Y_{\ell}+ o\left(1\right)\big)\left(\log_2 T \right)^{\ell+1} \,, \end{equation*} where $\mathbf Y_{\ell} = \int_0^{\infty} u^{\ell} ρ(u) du$. Here $ρ(u)$ is the Dickman function. We have $\mathbf Y_{\ell} > e^γ/(\ell + 1)$ and $ \log\, \mathbf Y_{\ell} = \left(1 + o\left(1\right) \right) \ell \log \ell$ when $ \ell \to \infty $, which significantly improves previous results in [17, 40]. Similar results are established for Dirichlet $L$-functions. On the other hand, when assuming the Riemann Hypothesis and the Generalized Riemann Hypothesis, we establish upper bounds for $ \left| ζ^{(\ell)}\left(1+it\right)\right| $ and $\left|L^{(\ell)}(1, χ) \right|$. Furthermore, when assuming the Granville-Soundararajan Conjecture is true, we establish the following asymptotic formulas $$\max_{ \substack{ χ\neq χ_0 \\ χ(\text{mod}\, q)}} \left|L^{(\ell)}(1, χ) \right| \sim \mathbf Y_{\ell}\left(\log_2 q\right)^{\ell+1},\,\, \quad \text{as}\,\quad q \to \infty,$$ where $q$ is prime and $\ell \in \mathbb{N}$ is given.
