论文标题
Lambert系列对数系列，Deninger函数$ r（z）$的衍生物和$ζ\ left的平均值定理（\ frac {1} {2} {2} {2} -it \ right \ right）ζ'\ left（\ frac {1}
Lambert series of logarithm, the derivative of Deninger's function $R(z)$ and a mean value theorem for $ζ\left(\frac{1}{2}-it\right)ζ'\left(\frac{1}{2}+it\right)$
论文作者
论文摘要
系列$ \ sum \ limits_ {n = 1}^{\ infty} \ displayStyle \ frac {\ log（n）} {e^{e^{ny} -1}，$ re $（re re $（y $ re $ $（y $ re $ $ $ re $）> 0 $，这是$ y $至$ y $ the y $至1/y $，获得的。该系列转变为包含$ψ_1（z）$的系列，这是Deninger函数$ r（z）$的派生。在获得转换过程中，$ψ_1（z）$的新重要属性是衍生的，对于两种可变量的mittag-leffler函数的第二个导数的新表示形式也是$ b = 1 $的第二个衍生物。我们的转换很容易给出$ \ sum \ limits_ {n = 1}^{\ infty} \ displayStyle \ frac {\ log（n）} {e^{e^{ny} -1} $ as y y y \ y \ to0 $。后者的应用是，它给出了$ \ displayStyle \ int_ {0}^{\ infty}}ζ}ζ\ left（\ frac {1} {2} {2} -it \ right）ζ' $δ\ to0 $。
An explicit transformation for the series $\sum\limits_{n=1}^{\infty}\displaystyle\frac{\log(n)}{e^{ny}-1},$ Re$(y)>0$, which takes $y$ to $1/y$, is obtained for the first time. This series transforms into a series containing $ψ_1(z)$, the derivative of Deninger's function $R(z)$. In the course of obtaining the transformation, new important properties of $ψ_1(z)$ are derived, as is a new representation for the second derivative of the two-variable Mittag-Leffler function $E_{2, b}(z)$ evaluated at $b=1$. Our transformation readily gives the complete asymptotic expansion of $\sum\limits_{n=1}^{\infty}\displaystyle\frac{\log(n)}{e^{ny}-1}$ as $y\to0$. An application of the latter is that it gives the asymptotic expansion of $ \displaystyle\int_{0}^{\infty}ζ\left(\frac{1}{2}-it\right)ζ'\left(\frac{1}{2}+it\right)e^{-δt}\, dt$ as $δ\to0$.
