Nobody knows whether Euler's constant is a fraction, and Hardy bet his chair
Euler's constant has been studied since 1734 and computed to vast numbers of digits, yet mathematicians still cannot prove it is irrational. David Hilbert called the question unapproachable, and G. H. Hardy is said to have promised his prestigious Oxford professorship to whoever could settle it.
The constant, written with the Greek letter gamma, measures a gap. Add up one, one half, one third and so on, and the total grows forever, a fact known for centuries before Euler. He showed that it grows at essentially the same rate as the natural logarithm, and the difference between the two settles down toward a fixed number as you go on. That limiting difference is Euler's constant.
Euler introduced it in a 1734 paper on harmonic progressions, calling it worthy of serious consideration. He first worked it out to 6 decimal places and, in 1781, to 16. The Italian Lorenzo Mascheroni tried for 32 places but got several digits wrong from the 20th onward, which is why his name is sometimes attached to it. Neither man used gamma; Euler wrote C or O, Mascheroni A or a, and Johann von Soldner H in 1809. The Greek letter came later, perhaps because of the link to the gamma function, and appears in work by Carl Anton Bretschneider in 1835 and Augustus De Morgan soon after.
Srinivasa Ramanujan published a paper on it in 1917, and one rapidly converging approximation is named the Ramanujan expansion after him. Shifting the logarithm by a half, rather than using the conventional form, makes such approximations home in far faster.
The constant turns up throughout number theory and analysis. It appears in formulas for the gamma function and the Riemann zeta function, in an inequality for Euler's totient function, in the average number of divisors of the numbers up to a given limit, in a conjecture about how often Mersenne primes occur, and in estimates of how efficient the Euclidean algorithm is.
Source: Euler's constant
