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In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers:
This also equals n times the inverse of the harmonic mean of these natural numbers.
The numbers n such that the numerator of the fully reduced expression for H_n is prime are
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.^{[1]}^{: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.
Bertrand's postulate entails that, except for the case n=1, the harmonic numbers are never integers.^{[2]}
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:
where B_k are the Bernoulli numbers.
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 example:
For positive integers p and q with p < q, we have:
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.
In 2002, Jeffrey Lagarias proved^{[5]} that the Riemann hypothesis is equivalent to the statement that
is true for every integer n ≥ 1 with strict inequality if n > 1; here σ(n) denotes the sum of the divisors of n.
The eigenvalues of the nonlocal problem
are given by \lambda = 2H_n, where by convention, H_0 = 0.
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 H_{n,0}= n
The special case of m = 1 is simply called a harmonic number and is frequently written without the superscript, as
Smallest natural number k such that k^{n} does not divide the denominator of generalized harmonic number H(k, n) nor the denominator of alternating generalized harmonic number H'(k, n) are
In the limit of n → ∞, the generalized harmonic number converges to the Riemann zeta function
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.
Some integrals of generalized harmonic are
and
Every generalized harmonic number of order m can be written as a function of harmonic of order m-1 using:
A generating function for the generalized harmonic numbers is
where \mathrm{Li}_m(z) 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 p,q>0 integer, and m>1 integer or not, we have from polygamma functions:
where \zeta(m) is the Riemann zeta function. The relevant recurrence relation is:
Some special values are:
The multiplication theorem applies to harmonic numbers. Using polygamma functions, we obtain
or, more generally,
For generalized harmonic numbers, we have
where \zeta(n) is the Riemann zeta function.
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
which is just the Newton's generalized binomial theorem. The interpolating function is in fact the digamma function
where \psi(x) is the digamma, and γ is the Euler-Mascheroni constant. The integration process may be repeated to obtain
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:
The next generalization was discussed by J. H. Conway and R. K. Guy in their 1995 book The Book of Numbers.^{[1]}^{:258} Let
Then the nth hyperharmonic number of order r (r>0) is defined recursively as
In special, H_n=H_n^{(1)}.
This article incorporates material from Harmonic number on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
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