# Harmonic number

In mathematics, the generalized harmonic number of order [itex]n[itex] is given by

[itex]H_{n,m}=\sum_{k=1}^n \frac{1}{k^m}.[itex]

The special case of [itex]m=1[itex] is simply called a harmonic number and is frequently written without the superscript, as

[itex]H_n= \sum_{k=1}^n \frac{1}{k}.[itex]

In the limit of [itex]n\rightarrow \infty[itex], the generalized harmonic number converges to the Riemann zeta function

[itex]\lim_{n\rightarrow \infty} H_{n,m} = \zeta(m)[itex]

The related sum [itex]\sum_{k=1}^n k^m[itex] occurs in the study of Bernoulli numbers.

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## Applications

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

[itex] \psi(n) = H_{n-1} - \gamma\, [itex]

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

[itex] \gamma = \lim_{n \rightarrow \infty}{\left(H_n - \ln(n)\right)} [itex]

and may be calculated from the formula:

[itex] H_n = \int_0^1 \frac{1 - x^n}{1 - x}\,dx [itex]

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:

[itex] \sigma(n) \le H_n + \ln(H_n)e^{H_n}[itex]

for every natural number n.

## References

• Art and Cultures
• Countries of the World (http://www.academickids.com/encyclopedia/index.php/Countries)
• Space and Astronomy