Criteria for irrationality of generalized Euler's constant

Abstract

We generalize Sondow's (ir)rationality criteria for Euler's constant and give necessary and sufficient conditions for irrationality of generalized Euler's constant g a ; as well as obtain new asymptotic formulas for computing g a : The proof is based on constructing linear forms in 1; g a and logarithms of rational numbers. r 2004 Elsevier Inc. All rights reserved.

Arithmetical properties of some series with logarithmic coefficients

“…Theorem 2.1 For j = 1, 2 and n → ∞ the following asymptotic formulas hold Note that series (6), (7) can also be given by the following double integrals (for the scheme of proof see [11])…”

“…First conditional irrationality criteria for Euler's constant were obtained by Sondow [10]. Further generalizations can be found in [7,13]. These criteria reduce the irrationality problem to checking if some effectively constructed sequence of linear forms in logarithms of natural numbers contains infinitely many terms with not very small fractional parts.…”

Fast convergences towards Euler-Mascheroni constant

“…We also refer here to the papers [1,2,[5][6][7][8][9][10][11][12][20][21][22][23][24][25][26][27], where important improvements of the speed of convergence of γ n were established. The complete asymptotic expansion of the sequence (γ n ) n≥1 is…”

On conditional irrationality measures for values of the digamma function

“…In [4] we gave irrationality criteria for the values of the digamma function (or the generalized Euler constant)…”

“…We then apply them to prove conditional irrationality measures for values of the digamma function γ α = − (α) (α) using a Diophantine approximation construction from [4].…”