Editing Harmonic number (section)

==Applications==
The harmonic numbers appear in several calculation formulas, such as the [[digamma function]]
<math display="block"> \psi(n) = H_{n-1} - \gamma.</math>
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 {{mvar|γ}} using the limit introduced earlier:
<math display="block"> \gamma = \lim_{n \rightarrow \infty}{\left(H_n - \ln(n)\right)}, </math>
although
<math display="block"> \gamma = \lim_{n \to \infty}{\left(H_n - \ln\left(n+\frac{1}{2}\right)\right)} </math>
converges more quickly.

In 2002, [[Jeffrey Lagarias]] proved<ref>{{cite journal |author=Jeffrey Lagarias |title=An Elementary Problem Equivalent to the Riemann Hypothesis |journal=Amer. Math. Monthly |volume=109 |issue=6 |year=2002 |pages=534–543 |arxiv=math.NT/0008177 |doi=10.2307/2695443|jstor=2695443 }}</ref> that the [[Riemann hypothesis]] is equivalent to the statement that
<math display="block"> \sigma(n) \le H_n + (\log H_n)e^{H_n},</math>
is true for every [[integer]] {{math|''n'' ≥ 1}} with strict inequality if {{math|''n'' > 1}}; here {{math|''σ''(''n'')}} denotes the [[divisor function|sum of the divisors]] of {{mvar|n}}.

The eigenvalues of the nonlocal problem on <math> L^2([-1,1])</math>
<math display="block"> \lambda \varphi(x) = \int_{-1}^{1} \frac{\varphi(x)-\varphi(y)}{|x-y|} \, dy </math>
are given by <math>\lambda = 2H_n</math>, where by convention <math>H_0 = 0</math>, and the corresponding eigenfunctions are given by the [[Legendre polynomials]] <math>\varphi(x) = P_n(x)</math>.<ref>{{cite journal |author=E.O. Tuck |title=Some methods for flows past blunt slender bodies |journal=J. Fluid Mech. |volume=18 |year=1964 |issue=4 |pages=619–635 |doi=10.1017/S0022112064000453|bibcode=1964JFM....18..619T |s2cid=123120978 }}</ref>