Euler–Mascheroni constant

Add 1, one half, one third, and so on. Compare that sum with the area under the curve one divided by x, from 1 to the last denominator. The sum stays ahead, but the difference approaches a fixed number: about 0.57721566. This is the Euler–Mascheroni constant. Can its exact value be a fraction of two integers? For the first two terms, the sum is 1.5 and the area is about 0.6931, leaving about 0.8069; this is an approximation step, not the final constant.[1] [2]

A reciprocal sum and the area under the same decreasing curve both grow without bound, but their difference approaches a finite number. Can that limiting number, the Euler–Mascheroni constant, be an exact fraction of integers? The sum uses the values at 1, 2, and later positive integers; the comparison area is the natural logarithm of the cutoff.

Write the harmonic number as $H_n=\sum_{k=1}^{n}1/k$. Then the definition is

$$\gamma=\lim_{n\to\infty}(H_n-\log n).$$

For example, $H_{10}=7381/2520$, about 2.928968, whereas $\log10$ is about 2.302585. Their difference is about 0.626383. Increasing the cutoff improves the limiting approximation. Knowing the limit exists and calculating many digits do not show whether there are integers $a,b$, with $b\ne0$, for which $\gamma=a/b$.[1] [2]

We do not know whether it is rational or irrational.[1] [2]

The checked result distinguishes γ from a transcendental combination involving it. A result about that combination does not establish the rationality, irrationality or transcendence of γ itself.[1]

The constant measures the lasting difference between adding values at separate integer points and using a smooth area to approximate them. Its exact arithmetic nature would tell us what kind of number this basic comparison produces.[2]

The constant also connects this sum-versus-area comparison to combinatorics. A uniformly random permutation rearranges labeled objects; following each object through the rearrangement divides them into cycles. The average number of cycles is $H_n$, so γ is also the limiting correction to its logarithmic growth. This is one reason to understand the exact constant rather than only a decimal approximation: the same number organizes both an analytic limit and a discrete counting problem.[2]