Wiktionary
n. (context number theory English) Any of a series of numbers formed from the sum of the reciprocals of consecutive natural numbers
Wikipedia
In mathematics, the -th harmonic number is the sum of the reciprocals of the first natural numbers:
$$H_n= 1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n} =\sum_{k=1}^n \frac{1}{k}.$$
Harmonic numbers are related to the harmonic mean in that the -th harmonic number is also times the reciprocal of the harmonic mean of the first positive integers.
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 harmonic numbers roughly approximate the natural logarithm function 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 most-valuable items is proportional to the -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 , the harmonic numbers are never integers.
In number theory, the harmonic numbers are the sums of the inverses of integers, forming the harmonic series. Harmonic number may also refer to:
- Harmonic, a periodic wave with a frequency that is an integral multiple of the frequency of another wave
- An integer not divisible by any prime other than 2 or 3, as defined by Philippe de Vitry
- Harmonic divisor numbers, also called Ore numbers or Ore's harmonic numbers, positive integers whose divisors have an integral harmonic mean