An Integral Representation of Logarithmic Derivative of Zeta Function

Question

How to show that: $$ \int_{0}^{\infty}\frac{1}{e^x-1}\left[\frac{12\,e^x}{(e^x-1)^2}-\frac{12}{x^2}+1\right]\,dx=\frac{5}{2}+\frac{\zeta'(2)}{\zeta(2)}-\frac{\zeta'(0)}{\zeta(0)}\tag{1} $$ Is it possible to split the integral into two integrals representing ${\zeta'}/{\zeta}(0)$ and ${\zeta'}/{\zeta}(2)$.

The logarithmic derivative of Riemann Zeta function is mainly defined by the infinte series: $$ \log'\left[\zeta(s)\right]=\frac{\zeta'}{\zeta}(s)=-\sum_{n=1}^{\infty}\frac{\Lambda(n)}{n^s} $$ Where $\Lambda(n)$ is Von Mangoldt function. ${\zeta'}/{\zeta}$ has some intresting idendities, like: $$ \frac{\zeta'}{\zeta}(s)+\frac{\zeta'}{\zeta}(1-s)=\log\pi-\frac{1}{2}\frac{\Gamma'}{\Gamma}\left(\frac{s}{2}\right)-\frac{1}{2}\frac{\Gamma'}{\Gamma}\left(\frac{1-s}{2}\right) $$ Does ${\zeta'}/{\zeta}$ has an integral representation? Thanks.

Answer 1

Let us consider the integral $$I_{1}\left(s\right)=\int_{0}^{\infty}\frac{x^{s-1}e^{x}}{\left(e^{x}-1\right)^{3}}dx,\,\mathrm{Re}\left(s\right)>3.$$ Then, by the dominated convergence theorem, we get $$I_{1}\left(s\right)=-\frac{1}{2}\sum_{k\geq2}k\left(k-1\right)\int_{0}^{\infty}x^{s-1}e^{-kx}dx$$ $$=-\frac{1}{2}\Gamma\left(s\right)\left(\zeta\left(s-2\right)-\zeta\left(s-1\right)\right).$$ In a similar fashion we have $$I_{2}\left(s\right)=\int_{0}^{\infty}\frac{x^{s-1}}{e^{x}-1}dx=\Gamma\left(s\right)\zeta\left(s\right),\mathrm{Re}\left(s\right)>1$$ so we can conclude that $$I_{3}\left(s\right)=\int_{0}^{\infty}\frac{x^{s-1}}{e^{x}-1}\left(\frac{12e^{x}}{\left(e^{x}-1\right)^{2}}-\frac{12}{x^{2}}+1\right)dx$$ $$=-\frac{12}{2}\Gamma\left(s\right)\left(\zeta\left(s-1\right)-\zeta\left(s-2\right)\right)-12\Gamma\left(s-2\right)\zeta\left(s-2\right)+\Gamma\left(s\right)\zeta\left(s\right),\mathrm{Re}\left(s\right)>3$$ but, using the asymptotic expansion of $\Gamma\left(s\right)$ at $s=0$ and $\zeta\left(s\right)$ at $s=1$ we can see that $I\left(s\right)$ is defined for $\mathrm{Re}\left(s\right)\geq1$ and $$\lim_{s\rightarrow1}-\frac{12}{2}\Gamma\left(s\right)\left(\zeta\left(s-1\right)-\zeta\left(s-2\right)\right)-12\Gamma\left(s-2\right)\zeta\left(s-2\right)+\Gamma\left(s\right)\zeta\left(s\right)$$ $$=\color{red}{\frac{5}{2}-\gamma+12\log\left(A\right)}$$ where $A$ is the Glaisher–Kinkelin constant and $\gamma$ is the Euler-Mascheroni constant, which is equivalet to your result.