A note on log-type GCD sums and derivatives of the Riemann zeta function

Abstract

In [Yan22a], we defined so-called ``log-type" GCD sums and proved the lower bounds $\Gamma^{(\ell)}_1(N) \gg_{\ell} \left(\log\log N\right)^{2+2\ell}$. We will establish the upper bounds $\Gamma^{(\ell)}_1(N)\ll_{\ell} \left(\log \log N\right)^{2+2\ell}$ in this note, which generalizes G\'{a}l's theorem on GCD sums (corresponding to the case $\ell = 0$). This result will be proved by two different methods. The first method is unconditional. We establish sharp upper bounds for spectral norms along $\alpha-$lines when $\alpha$ tends to $1$ with certain fast rates. As a corollary, we obtain upper bounds for log-type GCD sums. The second method is conditional. We prove that lower bounds for log-type GCD sums $\Gamma^{(\ell)}_1(N)$ can produce lower bounds for large values of derivatives of the Riemann zeta function on the 1-line. So from conditional upper bound for $\left| \zeta^{(\ell)}\left(1+ i t\right)\right|$, we obtain upper bounds for log-type GCD sums.