A Survey of Euler's Constant

Abstract

The mathematical constant y ? lim^oo (YH=i ? ln(rc)) ? 0.5772156..., known as Euler's constant, is not as well known as its cousins 7T, e, i, but is still important enough to warrant serious consideration in the circles of applied mathematics, calcu lus, and number theory. Some authors will occasionally refer to y as the Euler-Mascheroni constant, so named after the Italian geometer Lorenzo Mascheroni (1750-1800), who actually in troduced the symbol y for the constant (although there is controversy about this claim) and also computed, though with error, the first 32 digits [16, 34]. Sometimes one will find in older texts the symbols C (this was Euler's constant of integration) and A (also from Mascheroni) to represent the constant, but these notations seem to have disap peared in the modern era [27]. Our aim in this article is to present a survey of y that is both manageable by, and enlightening to, those who favor mathematics at the undergraduate level. To try and follow in the footsteps of the big boys it and e is quite a chore, but this brief historical description of y and colorful portfolio of applications and surprising appearances in a multitude of settings is both impressive and mathematically educational.