Harmonic Number - Applications

Applications

The harmonic numbers appear in several calculation formulas, such as the digamma function:

This relation is also frequently used to define the extension of the harmonic numbers to non-integer n. The harmonic numbers are also frequently used to define γ, using the limit introduced in the previous section, although

converges more quickly.

In 2002 Jeffrey Lagarias proved that the Riemann hypothesis is equivalent to the statement that

is true for every integer n ≥ 1 with strict inequality if n > 1; here σ(n) denotes the sum of the divisors of n.

The eigenvalues of the nonlocal problem

are given by, where by convention,