Harmonic number

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The harmonic number with (red line) with its asymptotic limit (blue line).

In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers:

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 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[1]: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 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]

Identities involving harmonic numbers[edit]

By definition, the harmonic numbers satisfy the recurrence relation

The harmonic numbers are connected to the Stirling numbers of the first kind:

The functions

satisfy the property

In particular

is an integral of the logarithmic function.

The harmonic numbers satisfy the series identity

Identities involving π[edit]

There are several infinite summations involving harmonic numbers and powers of π:[3]

Calculation[edit]

An integral representation given by Euler[4] is

The equality above is obvious by the simple algebraic identity

Using the simple integral transform x = 1−u, an elegant combinatorial expression for Hn is

The same representation can be produced by using the third Retkes identity by setting and using the fact that

The Taylor series for the harmonic numbers is

which comes from the Taylor series for the Digamma function.

Graph demonstrating a connection between harmonic numbers and the natural logarithm. The harmonic number Hn can be interpreted as a Riemann sum of the integral:

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

whose value is ln(n).

The values of the sequence Hn - ln(n) decrease monotonically towards the limit

where γ ≈ 0.5772156649 is the Euler–Mascheroni constant. The corresponding asymptotic expansion as n → +∞ is

where are the Bernoulli numbers.

Special values for fractional arguments[edit]

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 complex number x except the negative integers, 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:

Generating functions[edit]

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.

Applications[edit]

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:

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 , where by convention,

Generalization[edit]

Generalized harmonic numbers[edit]

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

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 kn does not divide the denominator of generalized harmonic number H(k, n) nor the denominator of alternating generalized harmonic number H'(k, n) are

77, 20, 94556602, 42, 444, 20, 104, 42, 76, 20, 77, 110, 3504, 20, 903, 42, 1107, 20, 104, 42, 77, 20, 2948, 110, 136, 20, 76, 42, 903, 20, 77, 42, 268, 20, 7004, 110, 1752, 20, 19203, 42, 77, 20, 104, 42, 76, 20, 370, 110, 1107, 20, ... (sequence A128670 in the OEIS)

In the limit of n → +∞, the generalized harmonic number converges to the Riemann zeta function

The related sum 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

where A is the Apéry's constant, i.e. ζ(3).

and

  for

Every generalized harmonic number of order m can be written as a function of harmonic of order m-1 using:

  for example:

A generating function for the generalized harmonic numbers is

where 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 integer, and integer or not, we have from polygamma functions:

where is the Riemann zeta function. The relevant recurrence relation is:

Some special values are:

where G is the Catalan's constant

Multiplication formulas[edit]

The multiplication theorem applies to harmonic numbers. Using polygamma functions, we obtain

or, more generally,

For generalized harmonic numbers, we have

where is the Riemann zeta function.

Generalization to the complex plane[edit]

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 the definition to the complex plane, other than the negative integers s for which the integral and sum diverge. 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 is the digamma, and γ is the Euler-Mascheroni constant. The integration process may be repeated to obtain

Relation to the Riemann zeta function[edit]

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:

Hyperharmonic numbers[edit]

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, .

See also[edit]

Notes[edit]

  1. ^ a b John H., Conway; Richard K., Guy (1995). The book of numbers. Copernicus. 
  2. ^ Ronald L., Graham; Donald E., Knuth; Oren, Patashnik (1994). Concrete Mathematics. Addison-Wesley. 
  3. ^ Sondow, Jonathan and Weisstein, Eric W. "Harmonic Number." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/HarmonicNumber.html
  4. ^ Sandifer, C. Edward (2007), How Euler Did It, MAA Spectrum, Mathematical Association of America, p. 206, ISBN 9780883855638 .
  5. ^ Jeffrey Lagarias (2002). "An Elementary Problem Equivalent to the Riemann Hypothesis". Amer. Math. Monthly 109: 534–543. arXiv:math.NT/0008177. doi:10.2307/2695443. 

References[edit]

External links[edit]

This article incorporates material from Harmonic number on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.