The harmonic numbers are the partial sums of the harmonic series. The harmonic number is the sum of the reciprocals of each positive integer up to . The first few harmonic numbers are as follows:
The harmonic number is the sum of the reciprocals of the first positive integers:
It can also be expressed with the recurrence relation
is a rational number. However,
There does not exist an integer such that is an integer.
Write Then is a sum of all the products of distinct integers between and inclusive. By Bertrand's postulate, there is a prime in the interval for Then divides exactly one of the integers between and inclusive, namely itself because So is a sum of all the products containing plus one that does not, which is not divisible by So but so cannot be an integer because its denominator, when it is reduced to lowest terms, will contain a factor of
The proof predicts that will contain a factor of in its denominator; indeed,
and it is clear that the denominator is divisible by but the numerator is not.
It is somewhat surprising that harmonic numbers have no upper bound. That is, for any real number , there exists a harmonic number that is greater than
The harmonic series diverges.
Suppose that the harmonic series converges, and consider the following series:
where the grouping symbols denote the ceiling function.
Each term in this sequence is positive and less than or equal to the corresponding term in the harmonic series:
Then if the harmonic series converges, this series converges as well.
However, this series does not converge. Grouping the like terms gives a repeated sum of
The fact that this series diverges is a contradiction. Therefore, the harmonic series diverges.
The harmonic numbers appear in expressions for integer values of the digamma function:
The harmonic numbers are used in the definition of the Euler-Mascheroni constant
The harmonic numbers are used in an elementary reformulation (due to Lagarias) of the Riemann hypothesis.
The Riemann hypothesis is equivalent to the statement
for all integers with strict inequality if where denotes the sum of the positive divisors of
An integral representation for given by Euler is
This is clear since
This can be used to find an alternating series representation of using the substitution
The generating function for the harmonic numbers has a relatively simple closed form
using the Maclaurin series for