On irrationality of Euler's constant and related asymptotic formulas

2023/08/03 by Suman, Shekhar
#FOS: Mathematics #General Mathematics (math.GM)

paper · doi:10.48550/arxiv.2312.00295

Abstract

By defining I_n:=∫₀¹∫₀¹ ((x(1-x)y(1-y))ⁿ)/((1-xy)(-log xy)) dx dy Sondow (see [2]) proved that I_n=\binom2nn γ+L_n-A_n We prove asymptotic formula for L_n and A_n as n→∞, L_n=\binom2nn(log ( (3n)/(2) ) +O ( (1)/(n) )) and A_n∼(4ⁿ)/(√(πn))(γ+ln\frac32+ln n) Using the sufficient condition for irrationality criteria of Euler's constant due to Sondow, we prove that γ is irrational.

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