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 the expressions of various special functions.
The associated harmonic series grows without limit, albeit very slowly, roughly approaching the natural logarithm function.:143 In 1737, Leonhard Euler used the divergence of this series to provide a new proof of the infinity of prime numbers. His work was extended into the complex plane by Bernhard Riemann in 1859, leading directly to the celebrated Riemann hypothesis about the distribution of prime numbers.
When the value of a large quantity of items has a Zipf's law distribution, the total value of the n most-valuable items is the n-th harmonic number. This leads to a variety of surprising conclusions in the Long Tail and the theory of network value.
- 1 Identities involving harmonic numbers
- 2 Calculation
- 3 Generating functions
- 4 Applications
- 5 Generalization
- 6 See also
- 7 Notes
- 8 References
- 9 External links
Identities involving harmonic numbers
By definition, the harmonic numbers satisfy the recurrence relation
They also satisfy the series identity
The harmonic numbers are connected to the Stirling numbers of the first kind:
satisfy the property
is a primitive of the logarithmic function.
The equality above is obvious by the simple algebraic identity
Using the simple integral transform x = 1−u, an elegant combinatorial expression for Hn is
The same representation can be produced by using the third Retkes identity by setting and using the fact that
whose value is ln(n).
The values of the sequence Hn - ln(n) decrease monotonically towards the limit
where are the Bernoulli numbers.
Special values for fractional arguments
There are the following special analytic values for fractional arguments between 0 and 1, given by the integral
More values may be generated from the recurrence relation
or from the reflection relation
For every x > 0, integer or not, we have:
Based on this, it can be shown that:
where γ is the Euler–Mascheroni constant or, more generally, for every n we have:
A generating function for the harmonic numbers is
where ln(z) is the natural logarithm. An exponential generating function is
where Ein(z) is the entire exponential integral. Note that
where Γ(0, z) is the incomplete gamma function.
The harmonic numbers appear in several calculation formulas, such as the digamma function
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
converges more quickly.
The eigenvalues of the nonlocal problem
are given by , where by convention,
Generalized harmonic numbers
The generalized harmonic number of order n of m is given by
The limit as n tends to infinity exists if m > 1.
Other notations occasionally used include
The special case of m = 0 gives
The special case of m = 1 is simply called a harmonic number and is frequently written without the superscript, as
In the limit of n → ∞, the generalized harmonic number converges to the Riemann zeta function
Some integrals of generalized harmonic are
- where A is the Apéry's constant, i.e. ζ(3).
Every generalized harmonic number of order m can be written as a function of harmonic of order m-1 using:
- for example:
A generating function for the generalized harmonic numbers is
where is the polylogarithm, and |z| < 1. The generating function given above for m = 1 is a special case of this formula.
Fractional argument for generalized harmonic numbers can be introduced as follows:
For every integer, and integer or not, we have from polygamma functions:
where is the Riemann zeta function. The relevant recurrence relation is:
Some special values are:
- where G is the Catalan's constant
Using polygamma functions, we obtain
or, more generally,
For generalized harmonic numbers, we have
where is the Riemann zeta function.
Generalization to the complex plane
Euler's integral formula for the harmonic numbers follows from the integral identity
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
where is the digamma, and γ is the Euler-Mascheroni constant. The integration process may be repeated to obtain
Relation to the Riemann zeta function
Some derivatives of fractional harmonic numbers are given by:
And using Maclaurin series, we have for x < 1:
For fractional arguments between 0 and 1, and for a > 1:
Then the nth hyperharmonic number of order r (r>0) is defined recursively as
In special, .
- John H., Conway; Richard K., Guy (1995). The book of numbers. Copernicus.
- Ronald L., Graham; Donald E., Knuth; Oren, Patashnik (1994). Concrete Mathematics. Addison-Wesley.
- Sandifer, C. Edward (2007), How Euler Did It, MAA Spectrum, Mathematical Association of America, p. 206, ISBN 9780883855638.
- Jeffrey Lagarias (2002). "An Elementary Problem Equivalent to the Riemann Hypothesis". Amer. Math. Monthly 109: 534–543. arXiv:math.NT/0008177. doi:10.2307/2695443.
- Arthur T. Benjamin, Gregory O. Preston, Jennifer J. Quinn (2002). "A Stirling Encounter with Harmonic Numbers". Mathematics Magazine 75 (2): 95–103. doi:10.2307/3219141.
- Donald Knuth (1997). "Section 1.2.7: Harmonic Numbers". The Art of Computer Programming. Volume 1: Fundamental Algorithms (Third ed.). Addison-Wesley. pp. 75–79. ISBN 0-201-89683-4.
- Ed Sandifer, How Euler Did It — Estimating the Basel problem (2003)
- Peter Paule, Carsten Schneider (2003). "Computer Proofs of a New Family of Harmonic Number Identities". Adv. in Appl. Math. 31 (2): 359–378. doi:10.1016/s0196-8858(03)00016-2.
- Wenchang Chu (2004). "A Binomial Coefficient Identity Associated with Beukers' Conjecture on Apery Numbers". The Electronic Journal of Combinatorics 11: N15.
- Ayhan Dil, István Mező (2008). "A Symmetric Algorithm for Hyperharmonic and Fibonacci Numbers". Applied Mathematics and Computation 206 (2): 942–951. doi:10.1016/j.amc.2008.10.013.
- Zoltán Retkes (2008). "An extension of the Hermite–Hadamard Inequality". Acta Sci. Math. (Szeged) 74: 95–106.