\documentclass[12pt]{article}
\usepackage{amsmath,amssymb,amsfonts}
\begin{document}
We prove that approximation formulas for the logarithms of some
infinite products, in particular, for Euler's constant $\gamma$,
log $\frac{4}{\pi}$ and log $\sigma$, where $\sigma$ in Somos's
quadratic recurrence constant, in terms of classical Legendre
polynomials and partial sums of their series expansions. We also
give conditional irrationality and linear independence criteria
for these numbers. The main tools are Euler-type integrals,
hypergeometric series, and Laplace method.
\end{document}