γ, the harmonic remainder

The harmonic numbers grow like a logarithm; γ is precisely what the comparison leaves behind:

$$\gamma \;=\; \lim_{n\to\infty}\Bigl(\sum_{k=1}^{n}\frac1k \;-\; \ln n\Bigr) \;=\; 0.5772156649\ldots$$

The limit exists for the pleasant reason: the difference $H_n - \ln n$ decreases at every step, since $\ln(1+1/n) > 1/(n+1)$, and stays positive, since $H_n > \ln(n+1)$ by comparing the sum with the integral it staircases over. Decreasing and bounded below — done. Euler introduced the constant in 1734 and eventually computed sixteen digits by hand, using the expansion that still does the work:1

1

Truncated Euler–Maclaurin. Records now use Brent–McMillan (1980); the digits run to the hundreds of billions.

$$H_n \;=\; \ln n + \gamma + \frac{1}{2n} - \frac{1}{12n^2} + \frac{1}{120n^4} - \cdots$$

Everywhere in the theory

γ is the analytic glue of multiplicative number theory. It is the derivative of the Gamma function at one, $\Gamma^{\prime}(1) = -\gamma$; it calibrates Mertens’ product over primes,

$$\prod_{p\le x}\Bigl(1-\frac1p\Bigr) \;\sim\; \frac{e^{-\gamma}}{\ln x},$$

and it sits in the second term of Dirichlet’s divisor count,

$$\sum_{n\le x} d(n) \;=\; x\ln x + (2\gamma-1)\,x + O(\sqrt{x}\,).$$

Wherever a sum over integers is traded for an integral, γ is the boundary toll.

What is not known

Whether γ is irrational. This is not for lack of pressure: the continued fraction has been computed past 475,000 partial quotients, which shows that if γ is a fraction $p/q$, then $q > 10^{244663}$.2 No rational number of any describable size is available to it — and yet no proof excludes them all. Three centuries separate the two-line definition from the first honest question about it, still open.

2

T. Papanikolaou’s computation (1997); see Havil, Gamma (2003). Irrationality of $e$ took one page; of $\pi$, a century more; of γ, counting.