Harmonic number
The harmonic numbers roughly approximate the natural logarithm function[2]: 143 and thus the associated harmonic series grows without limit, albeit slowly.
In 1737, Leonhard Euler used the divergence of the harmonic 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 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 Bertrand-Chebyshev theorem implies that, except for the case n = 1, the harmonic numbers are never integers.
[3] By definition, the harmonic numbers satisfy the recurrence relation
There are several infinite summations involving harmonic numbers and powers of π:[4][better source needed]
The nth harmonic number is about as large as the natural logarithm of n. The reason is that the sum is approximated by the integral
has zero asymptotic density, while Bing-Ling Wu and Yong-Gao Chen[11] proved that the number of elements of
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:
In 2002, Jeffrey Lagarias proved[12] 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 on
The special case m = 1 reduces to the usual harmonic number:
as n → ∞ is finite if m > 1, with the generalized harmonic number bounded by and converging to the Riemann zeta function
A fractional argument for generalized harmonic numbers can be introduced as follows: For every
Then the nth hyperharmonic number of order r (r>0) is defined recursively as
The Roman Harmonic numbers,[14] named after Steven Roman, were introduced by Daniel Loeb and Gian-Carlo Rota in the context of a generalization of umbral calculus with logarithms.
is Stirling numbers of the first kind generalized to negative first argument, and
In fact, these numbers were defined in a more general manner using Roman numbers and Roman factorials, that include negative values for
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.
There is an asymptotic formulation that gives the same result as the analytic continuation of the integral just described.
Then use the recursion relation Hn = Hn−1 + 1/n backwards m times, to unwind it to an approximation for Hx.
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.
There are the following special analytic values for fractional arguments between 0 and 1, given by the integral
Which are computed via Gauss's digamma theorem, which essentially states that for positive integers p and q with p < q
