Starting from n = 1, the sequence of harmonic numbers begins:
Harmonic numbers are related to the harmonic mean in that the n-th harmonic number is also n times the reciprocal of the harmonic mean of the first n positive integers.
Harmonic numbers have been studied since 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.
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 proportional to the n-th harmonic number. This leads to a variety of surprising conclusions regarding the long tail and the theory of network value.
The harmonic numbers have several interesting arithmetic properties. It is well-known that is an integer if and only if, a result often attributed to Taeisinger. Indeed, using 2-adic valuation, it is not difficult to prove that for the numerator of is an odd number while the denominator of is an even number. More precisely,
with some odd integers and .
As a consequence of Wolstenholme's theorem, for any prime number the numerator of is divisible by . Furthermore, Eisenstein proved that for all odd prime number it holds
In 1991, Eswarathasan and Levine defined as the set of all positive integers such that the numerator of is divisible by a prime number They proved that
for all prime numbers and they defined harmonic primes to be the primes such that has exactly 3 elements.
Eswarathasan and Levine also conjectured that is a finite set for all primes and that there are infinitely many harmonic primes. Boyd verified that is finite for all prime numbers up to except 83, 127, and 397; and he gave a heuristic suggesting that the density of the harmonic primes in the set of all primes should be . Sanna showed that has zero asymptotic density, while Bing-Ling Wu and Yong-Gao Chen proved that the number of elements of not exceeding is at most , for all .
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 earlier:
The generalized harmonic number of order m of n is given by
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 m, as
The limit as n → ∞ is finite if m > 1, with the generalized harmonic number bounded by and converging to the Riemann zeta function
The smallest natural number k such that kn does not divide the denominator of generalized harmonic number H(k, n) nor the denominator of alternating generalized harmonic number H′(k, n) is, for n=1, 2, ... :
are an integral and a series representation for a function that interpolates the harmonic numbers and, via analytic continuation, extends the definition to the complex plane other than the negative integers x. The interpolating function is in fact closely related to the digamma function
When seeking to approximate Hx for a complex number x, it is effective to first compute Hm for some large integer m. Use that to approximate a value for Hm+x and then use the recursion relation Hn = Hn−1 + 1/n backwards m times, to unwind it to an approximation for Hx. Furthermore, this approximation is exact in the limit as m goes to infinity.
Specifically, for a fixed integer n, it is the case that
If n is not an integer then it is not possible to say whether this equation is true because we have not yet (in this section) defined harmonic numbers for non-integers. However, we do get a unique extension of the harmonic numbers to the non-integers by insisting that this equation continue to hold when the arbitrary integer n is replaced by an arbitrary complex number x.
Swapping the order of the two sides of this equation and then subtracting them from Hx gives
This infinite series converges for all complex numbers x except the negative integers, which fail because trying to use the recursion relation Hn = Hn−1 + 1/n backwards through the value n = 0 involves a division by zero. By this construction, the function that defines the harmonic number for complex values is the unique function that simultaneously satisfies (1) H0 = 0, (2) Hx = Hx−1 + 1/x for all complex numbers x except the non-positive integers, and (3) limm→+∞ (Hm+x − Hm) = 0 for all complex values x.
Note that this last formula can be used to show that:
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