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Linear independence measures for Chowla--Selberg periods

Wadim Zudilin

Abstract

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

Linear independence measures for Chowla--Selberg periods

Abstract

We use simultaneous Padé approximations to hypergeometric functions to estimate from below linear forms in , , and with integral coefficients, for certain choices of positive integer and negative integer , where is (the square of) a Chowla--Selberg period attached to the imaginary quadratic field .

Paper Structure

This paper contains 1 section, 1 theorem, 11 equations.

Table of Contents

  1. Appendix

Key Result

Theorem 1

For $(D,d)\in\{(-148,1),(-232,2),(-267,3),(-163,10005)\}$, define respectively; these are the estimates for the corresponding irrationality measures of $\pi\sqrt d$ computed in Zu05.The class numbers of the quadratic fields $\mathbb Q(\sqrt{-148}),\mathbb Q(\sqrt{-232}),\mathbb Q(\sqrt{-267})$ are equal to 2, while the class number of $\mathbb Q(\sqrt{-163})$ is 1

Theorems & Definitions (1)

  • Theorem 1