Harmonic number

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The term harmonic number has multiple meanings. For other meanings, see harmonic number (disambiguation).

The harmonic number

H n,1

with $n=\lfloor{x}\rfloor$ (red line) with its asymptotic limit

γ + ln[x]

(blue line).

In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers:

$H_n= 1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}$

$=\sum_{k=1}^n \frac{1}{k}.$

This also equals n times the inverse of the harmonic mean of these natural numbers.

Harmonic numbers were studied in antiquity and are important in various branches of number theory. They are sometimes loosely termed harmonic series, are closely related to the Riemann zeta function, and appear in various expressions for various special functions.

1 Calculation
2 Special values for fractional arguments
3 Generating functions
4 Applications
5 Generalization
- 5.1 Generalized harmonic numbers
- 5.2 Generalization to the complex plane
6 References

[edit] Calculation

An integral representation is given by Euler:

$H_n = \int_0^1 \frac{1 - x^n}{1 - x}\,dx.$

This representation can be easily shown to satisfy the recurrence relation by the formula

$\int_0^1 x^n\,dx = \frac{1}{n + 1},$

and then

$x^{n} + \frac{1 - x^n}{1 - x} = \frac{1 - x^{n+1}}{1 - x}$

inside the integral.

H_n grows about as fast as the natural logarithm of n. The reason is that the sum is approximated by the integral

$\int_1^n {1 \over x}\, dx$

whose value is ln(n). More precisely, we have the limit:

$\lim_{n \to \infty} H_n - \ln(n) = \gamma$

(where γ is the Euler-Mascheroni constant $0.5772156649\dots$ ), and the corresponding asymptotic expansion:

$H_n = \gamma + \ln{n} + \frac{1}{2}n^{-1} - \frac{1}{12}n^{-2} + \frac{1}{120}n^{-4} + \mathcal{O}(n^{-6})$

[edit] Special values for fractional arguments

There are the following special analytic values for fractional arguments between 0 and 1, given by the integral

$H_\alpha = \int_0^1\frac{1-x^\alpha}{1-x}\,dx$

More may be generated from the recurrence relation $H_\alpha = H_{\alpha-1}+\frac{1}{\alpha}$ .

$H_{1/2} = 2 -2\ln{2}\,$

$H_{1/3} = 3-\frac{\pi}{2\sqrt{3}} -\frac{3}{2}\ln{3}$

$H_{1/4} = 4-\frac{\pi}{2} - 3\ln{2}$

$H_{1/6} = 6-\frac{\pi}{2}\sqrt{3} -2\ln{2} -\frac{3}{2}\ln(3)$

$H_{1/8} = 8-\frac{\pi}{2} - 4\ln{2} - \frac{1}{\sqrt{2}} \left\{\pi + \ln(2 + \sqrt{2}) - \ln(2 - \sqrt{2})\right\}$

[edit] Generating functions

A generating function for the harmonic numbers is

$\sum_{n=1}^\infty z^n H_n = \frac {-\ln(1-z)}{1-z},$

where $ln(z)$ is the natural logarithm. An exponential generating function is

$\sum_{n=1}^\infty \frac {z^n}{n!} H_n = -e^z \sum_{k=1}^\infty \frac{1}{k} \frac {(-z)^k}{k!} = e^z \mbox {Ein}(z)$

where $Ein(z)$ is the entire exponential integral. Note that

$\mbox {Ein}(z) = \mbox{E}_1(z) + \gamma + \ln z = \Gamma (0,z) + \gamma + \ln z\,$

where $Γ(0, z)$ is the incomplete gamma function.

[edit] Applications

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

$\psi(n) = H_{n-1} - \gamma.\,$

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

$\gamma = \lim_{n \rightarrow \infty}{\left(H_n - \ln\left(n+{1 \over 2}\right)\right)}$

converges more quickly.

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

$\sigma(n) \le H_n + \ln(H_n)e^{H_n},$

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

[edit] Generalization

[edit] Generalized harmonic numbers

The generalized harmonic number of order $n$ of m is given by

$H_{n,m}=\sum_{k=1}^n \frac{1}{k^m}.$

Note that the limit as n tends to infinity exists if $m > 1$ .

Other notations occasionally used include

$H_{n,m}= H_n^{(m)} = H_m(n).$

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

$H_n= \sum_{k=1}^n \frac{1}{k}.$

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

$\lim_{n\rightarrow \infty} H_{n,m} = \zeta(m).$

The related sum $\sum_{k=1}^n k^m$ occurs in the study of Bernoulli numbers; the harmonic numbers also appear in the study of Stirling numbers.

A generating function for the generalized harmonic numbers is

$\sum_{n=1}^\infty z^n H_{n,m} = \frac {\mbox{Li}_m(z)}{1-z},$

where $Li m (z)$ is the polylogarithm, and $| z | < 1$ . The generating function given above for $m = 1$ is a special case of this formula.

[edit] Generalization to the complex plane

Euler's integral formula for the harmonic numbers follows from the integral identity

$\int_a^1 \frac {1-x^s}{1-x} dx = - \sum_{k=1}^\infty \frac {1}{k} {s \choose k} (a-1)^k$

which holds for general complex-valued s, for the suitably extended binomial coefficients. By choosing a=0, this formula gives both an integral and a series representation for a function that interpolates the harmonic numbers and extends a definition to the complex plane. This integral relation is easily derived by manipulating the Newton series

$\sum_{k=0}^\infty {s \choose k} (-x)^k = (1-x)^s,$

which is just the Newton's generalized binomial theorem. The interpolating function is in fact just the digamma function:

$\psi(s+1)+\gamma = \int_0^1 \frac {1-x^s}{1-x} dx$

where $ψ(x)$ is the digamma, and $γ$ is the Euler-Mascheroni constant. The integration process may be repeated to obtain

$H_{s,2}=-\sum_{k=1}^\infty \frac {(-1)^k}{k} {s \choose k} H_k.$

[edit] References

Arthur T. Benjamin, Gregory O. Preston, Jennifer J. Quinn, A Stirling Encounter with Harmonic Numbers, (2002) Mathematics Magazine, 75 (2) pp 95-103.
Donald Knuth. The Art of Computer Programming, Volume 1: Fundamental Algorithms, Third Edition. Addison-Wesley, 1997. ISBN 0-201-89683-4. Section 1.2.7: Harmonic Numbers, pp.75–79.
Ed Sandifer, How Euler Did It -- Estimating the Basel problem (2003)
Eric W. Weisstein, Harmonic Number at MathWorld.
Peter Paule and Carsten Schneider, Computer Proofs of a New Family of Harmonic Number Identities, (2003) Adv. in Appl. Math. 31(2), pp. 359-378.
Wenchang CHU, A Binomial Coefficient Identity Associated with Beukers' Conjecture on Apery Numbers, (2004) The Electronic Journal of Combinatorics, 11, #N15.

This article incorporates material from Harmonic number on PlanetMath, which is licensed under the GFDL.

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