Linear independence measures for Chowla--Selberg periods

TL;DR

This paper employs simultaneous Padé approximations of hypergeometric functions to derive lower bounds for linear forms involving fundamental constants related to imaginary quadratic fields.

Contribution

It introduces a novel approach using Padé approximations to estimate linear independence measures for Chowla--Selberg periods and related constants.

Findings

Established explicit lower bounds for linear forms in key constants.

Extended methods to specific imaginary quadratic fields.

Provided new insights into the arithmetic nature of Chowla--Selberg periods.

Abstract

We use simultaneous Pad\'e approximations to $_3F_2$ hypergeometric functions to estimate from below linear forms in $1$, $\pi\sqrt d$, $\Omega_D/\pi$ and $\pi/\Omega_D$ with integral coefficients, for certain choices of positive integer $d$ and negative integer $D$, where $\Omega_D$ is (the square of) a Chowla--Selberg period attached to the imaginary quadratic field $Q(\sqrt{D})$.