Harmonic number
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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.
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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
- How Euler Did It -- Estimating the Basel problem (http://www.maa.org/editorial/euler/How%20Euler%20Did%20It%2002%20Estimating%20the%20Basel%20Problem.pdf)
- Template:Journal reference
See also
External links
- Harmonic Number -- from MathWorld (http://mathworld.wolfram.com/HarmonicNumber.html)it:Numero armonico generalizzato