论文标题
关于riemann zeta函数的log-type gcd和衍生物的注释
A note on log-type GCD sums and derivatives of the Riemann zeta function
论文作者
论文摘要
在[yan22a]中，我们定义了所谓的``log-type''gcd总和，并证明了下限$γ^{（\ ell）} _ 1（n）\ gg _ {\ ell} \ loge（\ log log n \ n \ right） $γ^{（\ ell）} _ 1（n）\ ll _ {\ ell} \ feust（\ log \ log \ log n \ right）^{2+2 \ 2 \ ell} $在gcd sum上概括了gcd suhl n.当$α$的频谱规范的上限趋于$ 1 $，以某些快速汇率为$ $。 zeta功能在1线上。
In [Yan22a], we defined so-called ``log-type" GCD sums and proved the lower bounds $Γ^{(\ell)}_1(N) \gg_{\ell} \left(\log\log N\right)^{2+2\ell}$. We will establish the upper bounds $Γ^{(\ell)}_1(N)\ll_{\ell} \left(\log \log N\right)^{2+2\ell}$ in this note, which generalizes Gál'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 $α-$lines when $α$ 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 $Γ^{(\ell)}_1(N)$ can produce lower bounds for large values of derivatives of the Riemann zeta function on the 1-line. So from conditional upper bound for $\left| ζ^{(\ell)}\left(1+ i t\right)\right|$, we obtain upper bounds for log-type GCD sums.
