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The harmonic number with (red line) with its asymptotic limit (blue line) where is the Euler–Mascheroni constant.

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.

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 regarding 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]

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

Contents

Identities involving harmonic numbersEdit

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 πEdit

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

 
 
 
 

CalculationEdit

An integral representation given by Euler[4] is

 

The equality above is straightforward by the simple algebraic identity

 

Using the substitution x = 1−u, another expression for Hn is

 
 
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.

Generating functionsEdit

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.

Arithmetic propertiesEdit

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.[5] 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[6] proved that for all odd prime number   it holds

 

where   is a Fermat quotient, with the consequence that   divides the numerator of   if and only if   is a Wieferich prime. In 1991, Eswarathasan and Levine[7] 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 called harmonic primes the primes   such that   has exactly 3 elements.

Eswarathasan and Levine also conjectured that   is a finite set all primes number  , and that there are infinitely many harmonic primes. Boyd[8] verified that   is finite for all prime numbers up to  , but 83, 127, and 397; and he gave an heuristic suggesting that the relatively density of the harmonic primes in the set of all primes should be  . Sanna[9] showed that   has zero asymptotic density, while Bing-Ling Wu and Yong-Gao Chen[10] proved that the number of elements of   not exceeding   is at most  , for all  .

ApplicationsEdit

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[11] 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,  

GeneralizationsEdit

Generalized harmonic numbersEdit

The generalized harmonic number of order n of m is given by

 

The limit as n tends to infinity is finite 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 m, as

 

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, ... :

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 → +∞ for m > 1, 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 {{math1=|z| < 1}}. The generating function given above for m = 1 is a special case of this formula.

A 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 Catalan's constant
 
 
 
 
 

Multiplication formulasEdit

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.

Hyperharmonic numbersEdit

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 particular,   is the ordinary harmonic number  .

Harmonic numbers for real and complex valuesEdit

The formulae given above,

 

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

 

where ψ(x) is the digamma, and γ is the Euler-Mascheroni constant. The integration process may be repeated to obtain

 

The Taylor series for the harmonic numbers is

 

which comes from the Taylor series for the digamma function.

Alternative, asymptotic formulationEdit

When seeking to approximate Hx for a complex number x it turns out that it is effective to first compute Hm for some large integer m, then 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 every integer n, we have that

 

and we can ask that the formula be obeyed if the arbitrary integer n is replaced by an arbitrary complex number x

 

Adding Hx to both sides gives

 

This last expression for Hx is well defined for any complex number 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 construction, the function Hx is the unique function of x for which (1) H0 = 0, (2) Hx = Hx−1 + 1/x for all complex values x except the non-positive integers, and (3) limm→+∞ (Hm+xHm) = 0 for all complex values x.

Based on this last formula, it can be shown that:

 

where γ is the Euler–Mascheroni constant or, more generally, for every n we have:

 

Special values for fractional argumentsEdit

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:

 

Relation to the Riemann zeta functionEdit

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:

 
 
 

See alsoEdit

NotesEdit

  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. ^ Weisstein, Eric W. (2003). CRC Concise Encyclopedia of Mathematics. Boca Raton, FL: Chapman & Hall/CRC. p. 3115. ISBN 1-58488-347-2. 
  6. ^ Eisenstein, Ferdinand Gotthold Max (1850). "Eine neue Gattung zahlentheoretischer Funktionen, welche von zwei Elementen ahhängen und durch gewisse lineare Funktional-Gleichungen definirt werden". Berichte Königl. Preuβ. Akad. Wiss. Berlin. 15: 36–42. 
  7. ^ Eswarathasan, Arulappah; Levine, Eugene (1991). "p-integral harmonic sums". Discrete Mathematics. 91: 249–257. doi:10.1016/0012-365X(90)90234-9. 
  8. ^ Boyd, David W. (1994). "A p-adic study of the partial sums of the harmonic series". Experimental Mathematics. 3: 287–302. 
  9. ^ Sanna, Carlo (2016). "On the p-adic valuation of harmonic numbers". Journal of Number Theory. 166: 41–46. 
  10. ^ Chen, Yong-Gao; Wu, Bing-Ling (2017). "On certain properties of harmonic numbers". Journal of Number Theory. 175: 66–86. 
  11. ^ 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. 

ReferencesEdit

External linksEdit

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