Harmonic number

From Academic Kids

In mathematics, the generalized harmonic number of order <math>n<math> is given by

<math>H_{n,m}=\sum_{k=1}^n \frac{1}{k^m}.<math>

The special case of <math>m=1<math> is simply called a harmonic number and is frequently written without the superscript, as

<math>H_n= \sum_{k=1}^n \frac{1}{k}.<math>

In the limit of <math>n\rightarrow \infty<math>, the generalized harmonic number converges to the Riemann zeta function

<math>\lim_{n\rightarrow \infty} H_{n,m} = \zeta(m)<math>

The related sum <math>\sum_{k=1}^n k^m<math> occurs in the study of Bernoulli numbers.

Contents

Applications

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

<math> \psi(n) = H_{n-1} - \gamma\, <math>

where γ is the Euler-Mascheroni constant The harmonic numbers are also part of the definition of γ,

<math> \gamma = \lim_{n \rightarrow \infty}{\left(H_n - \ln(n)\right)} <math>

and may be calculated from the formula:

<math> H_n = \int_0^1 \frac{1 - x^n}{1 - x}\,dx <math>

due to Euler

Riemann hypothesis

Jeffrey Lagarias connected the harmonic numbers with the Riemann hypothesis in 2001 by proving that the Riemann hypothesis is equivalent with the statement:

<math> \sigma(n) \le H_n + \ln(H_n)e^{H_n}<math>

for every natural number n.

References

See also

External links

nl:Harmonisch getal

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