Lambert series of logarithm, the derivative of Deninger's function \(R(z)\), and a mean value theorem for \(\zeta \left (\frac{1}{2}-it\right)\zeta '\left (\frac{1}{2}+it\right)\) (Q6633509)

Lambert series are defined by \N\[\N\sum_{n\geq1}a(n)\frac{q^{n}}{1-q^{n}}=\sum_{n\geq1}a(n)\frac{1}{e^{ny}-1}=\sum_{n\geq1}(1*a)(n)e^{-ny},\N\]\Nwhere \(q=e^{-y}\) with \(\Re( y)>0\) and \(a(n)\) is an arithmetic function with \(\sum_{d\mid n}a(d)=(1*a)(n)\) as the Dirichlet convolution.\N\NIn this paper under review, the authors give in Theorem 1.1 an explicit formula for the Lambert series of logarithm \(\sum_{n\geq1}\log(n)\frac{1}{e^{ny}-1}\) in terms of the function \(\psi_{1}(z)\) for all \(z\in{\mathbb D:=\mathbb {C}\backslash_{\{x\in{\mathbb{R}/x\leq0\}}}}\) which is defined by \N\[\N\psi_{1}(z):=-\gamma_{1}-\frac{\log z}{z}-\sum_{n\geq1}\left(\frac{\log(n+z)}{n+z}- \frac{\log n}{n+z}\right)\N\]\Nwith \(\gamma_{1}\) is the first Stieltjes constant. In Theorem 1.2, they state an asymptotic formula for the above Lambert series of logarithm which has an application in the theory of moments (see Theorem 1.3). Actually, as \(\delta\to0\),\ \(|\arg(\delta)|<\pi/2\), one has \N\begin{align*} \N\int_{0}^{\infty}\zeta\left(\frac{1}{2}-it\right) \zeta'\left(\frac{1}{2}+it\right)e^{-\delta t}dt&=&-\frac{1}{4\sin(\delta/2)}\left(\log^{2}(2\pi\delta)+\frac{\pi^{2}}{\delta}-\gamma^{2}\right)\\ &&+\sum_{k=0}^{2m-2}(d_{k}+d'_{k}\log(\delta))\delta^{k} +O\left(\delta^{2m-1}\log(\delta)\right), \N\end{align*}\N where \(d_{k}\) and \(d'_{k}\) are effectively computable constants and the constant implied by the big-\(O\) depends on \(m\).