Extreme values of derivatives of the Riemann zeta function, log-type GCD sums, and spectral norms

Abstract: First I will recall the research on greatest common divisor (GCD) sums and extreme values of the Riemann zeta function. The motivation for the study and the connection between the two problems will be discussed. Then I will explain how to establish lower bounds for maximums of $|\zeta^{(\ell)}\left(\sigma+it\right)|$ when $\sigma \in [\frac{1}{2}, 1]$, $\ell \in \N$. One of my results states that as $T \to \infty$, uniformly for all positive integers $\ell \leqslant (\log_3 T) / (\log_4 T)$, we have $ \max_{T\leqslant t\leqslant 2T}\left|\zeta^{(\ell)}\left(1+it\right)\right| \geqslant \left(\mathbf Y_{\ell}+ o\left(1\right)\right)\left(\log_2 T \right)^{\ell+1} $, where $\mathbf Y_{\ell} = \int_0^{\infty} u^{\ell} \rho (u) du$, and $\rho(u)$ denotes the Dickman function. This generalizes results of Bohr-Landau and Littlewood on $\left|\zeta\left(1+it\right)\right|$ in 1910s. The tools are Soundararajan's resonance methods and ingredients are certain combinatorial optimization problems. On the other hand, assuming the Riemann hypothesis, we have $|\zeta^{(\ell)}\left(1+it\right)| \ll_{\ell}\left(\log \log t\right)^{\ell+1}$. Then I will talk on the log-type GCD sums $\Gamma^{(\ell)}_{\sigma}(N)$, which I define it as $\Gamma_{\sigma}^{(\ell)}(N):\,= \sup_{|\mathcal{M}| = N} \frac{1}{N}\sum_{m, n\in \mathcal{M}} \frac{(m,n)^{\sigma}}{[m,n]^{\sigma}}\log^{\ell} \left(\frac{m}{(m,n)}\right)\log^{\ell}\left(\frac{n}{(m,n)}\right),$ where the supremum is taken over all subsets $\mathcal{M} \subset \mathbb N$ with size $N$. I will explain how $\Gamma^{(\ell)}_{\sigma}(N)$ can be related to $|\zeta^{(\ell)}(1+it)|$ and how to prove that $\left(\log\log N\right)^{2+2\ell} \ll _{\ell}\Gamma^{(\ell)}_1(N)\ll_{\ell} \left(\log \log N\right)^{2+2\ell}$, which generalizes Gál's theorem (corresponding to the case $\ell = 0$). The lower bounds could be used to produce large values of $|\zeta^{(\ell)}\left(1+it\right)|$. Using a random model for the zeta function via methods of Lewko-Radziwiłł, upper bounds for spectral norms on $\alpha$-line are established, when $\alpha \to 1^{-}$ with certain fast rates. As a corollary, upper bounds of correct order of the log-type GCD sums are established.

number theory

Audience: researchers in the discipline

Combinatorial and additive number theory (CANT 2022)