Expression of Farhi's integral in terms of known mathematical constants

Abstract

In an interesting article entitled "A curious formula related to the Euler Gamma function", Bakir Farhi posed the open question of whether it was possible to obtain an expression of $$ \boldsymbol{\eta}=2\int_0^1\log\Gamma(x)\,\cdot\sin(2\pi x)\,\mathrm{d}x=0.7687478924\dots $$ in terms of the known mathematical constants as $\pi$, $\log\pi$, $\log 2$, $\Gamma(1/4)$, $e$, etc. In the present work, we show that $$ \boldsymbol{\eta}=\frac{1}{\pi}\left[\gamma+\log(2\pi)\right], $$ where $\gamma$ is the usual Euler-Mascheroni constant, and provide two different proofs, the first one involving the Glaisher-Kinkelin constant, and the second one based on the Malmst\'en integral representation of $\log\Gamma(x)$. The resulting formula can also be obtained directly from the knowledge of the Fourier series expansion of $\log\Gamma(x)$.