# Harmonic number

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 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]}

n |
Harmonic number, H
_{n} | |||
---|---|---|---|---|

expressed as a fraction | decimal | relative size | ||

1 | 1 | 1 | ||

2 | 3 | /2 | 1.5 | |

3 | 11 | /6 | ~1.83333 | |

4 | 25 | /12 | ~2.08333 | |

5 | 137 | /60 | ~2.28333 | |

6 | 49 | /20 | 2.45 | |

7 | 363 | /140 | ~2.59286 | |

8 | 761 | /280 | ~2.71786 | |

9 | 7 129 | /2 520 | ~2.82897 | |

10 | 7 381 | /2 520 | ~2.92897 | |

11 | 83 711 | /27 720 | ~3.01988 | |

12 | 86 021 | /27 720 | ~3.10321 | |

13 | 1 145 993 | /360 360 | ~3.18013 | |

14 | 1 171 733 | /360 360 | ~3.25156 | |

15 | 1 195 757 | /360 360 | ~3.31823 | |

16 | 2 436 559 | /720 720 | ~3.38073 | |

17 | 42 142 223 | /12 252 240 | ~3.43955 | |

18 | 14 274 301 | /4 084 080 | ~3.49511 | |

19 | 275 295 799 | /77 597 520 | ~3.54774 | |

20 | 55 835 135 | /15 519 504 | ~3.59774 | |

21 | 18 858 053 | /5 173 168 | ~3.64536 | |

22 | 19 093 197 | /5 173 168 | ~3.69081 | |

23 | 444 316 699 | /118 982 864 | ~3.73429 | |

24 | 1 347 822 955 | /356 948 592 | ~3.77596 | |

25 | 34 052 522 467 | /8 923 714 800 | ~3.81596 | |

26 | 34 395 742 267 | /8 923 714 800 | ~3.85442 | |

27 | 312 536 252 003 | /80 313 433 200 | ~3.89146 | |

28 | 315 404 588 903 | /80 313 433 200 | ~3.92717 | |

29 | 9 227 046 511 387 | /2 329 089 562 800 | ~3.96165 | |

30 | 9 304 682 830 147 | /2 329 089 562 800 | ~3.99499 | |

31 | 290 774 257 297 357 | /72 201 776 446 800 | ~4.02725 | |

32 | 586 061 125 622 639 | /144 403 552 893 600 | ~4.05850 | |

33 | 53 676 090 078 349 | /13 127 595 717 600 | ~4.08880 | |

34 | 54 062 195 834 749 | /13 127 595 717 600 | ~4.11821 | |

35 | 54 437 269 998 109 | /13 127 595 717 600 | ~4.14678 | |

36 | 54 801 925 434 709 | /13 127 595 717 600 | ~4.17456 | |

37 | 2 040 798 836 801 833 | /485 721 041 551 200 | ~4.20159 | |

38 | 2 053 580 969 474 233 | /485 721 041 551 200 | ~4.22790 | |

39 | 2 066 035 355 155 033 | /485 721 041 551 200 | ~4.25354 | |

40 | 2 078 178 381 193 813 | /485 721 041 551 200 | ~4.27854 |

## Contents

## 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 straightforward by the simple algebraic identity

Using the substitution *x* = 1−*u*, another expression for *H _{n}* is

A closed form expression for *H _{n}* is

where

The *n*th 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 *H*_{n} - 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 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.

## Arithmetic properties[edit]

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 .

## 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^{[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,

## Generalizations[edit]

### Generalized harmonic numbers[edit]

The **generalized harmonic number** of order *m* of *n* 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 *k ^{n}* 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 as *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 numbers are

and

- where
*A*is 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.

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 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.

### 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 particular, is the ordinary harmonic number .

## Harmonic numbers for real and complex values[edit]

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 formulation[edit]

When seeking to approximate *H*_{x} for a complex number *x* it turns out that it is effective to first compute *H*_{m} for some large integer *m*, then use that to approximate a value for *H*_{m+x}, and then use the recursion relation *H*_{n} = *H*_{n−1} + 1/*n* backwards *m* times, to unwind it to an approximation for *H*_{x}. 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 *H*_{x} to both sides gives

This last expression for *H*_{x} is well defined for any complex number *x* except the negative integers, which fail because trying to use the recursion relation *H*_{n} = *H*_{n−1} + 1/*n* backwards through the value *n* = 0 involves a division by zero. By construction, the function *H*_{x} is the unique function of *x* for which (1) *H*_{0} = 0, (2) *H*_{x} = *H*_{x−1} + 1/*x* for all complex values *x* except the non-positive integers, and (3) lim_{m→+∞} (*H*_{m+x} − *H*_{m}) = 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 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:

### 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:

## See also[edit]

- Watterson estimator
- Tajima's D
- Coupon collector's problem
- Jeep problem
- Riemann zeta function
- List of sums of reciprocals

## Notes[edit]

- ^
^{a}^{b}John H., Conway; Richard K., Guy (1995).*The book of numbers*. Copernicus. **^**Ronald L., Graham; Donald E., Knuth; Oren, Patashnik (1994).*Concrete Mathematics*. Addison-Wesley.**^**Sondow, Jonathan and Weisstein, Eric W. "Harmonic Number." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/HarmonicNumber.html**^**Sandifer, C. Edward (2007),*How Euler Did It*, MAA Spectrum, Mathematical Association of America, p. 206, ISBN 9780883855638.**^**Weisstein, Eric W. (2003).*CRC Concise Encyclopedia of Mathematics*. Boca Raton, FL: Chapman & Hall/CRC. p. 3115. ISBN 1-58488-347-2.**^**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.**^**Eswarathasan, Arulappah; Levine, Eugene (1991). "p-integral harmonic sums".*Discrete Mathematics*.**91**: 249–257. doi:10.1016/0012-365X(90)90234-9.**^**Boyd, David W. (1994). "A p-adic study of the partial sums of the harmonic series".*Experimental Mathematics*.**3**: 287–302. doi:10.1080/10586458.1994.10504298.**^**Sanna, Carlo (2016). "On the p-adic valuation of harmonic numbers".*Journal of Number Theory*.**166**: 41–46. doi:10.1016/j.jnt.2016.02.020.**^**Chen, Yong-Gao; Wu, Bing-Ling (2017). "On certain properties of harmonic numbers".*Journal of Number Theory*.**175**: 66–86. doi:10.1016/j.jnt.2016.11.027.**^**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]

- Arthur T. Benjamin; Gregory O. Preston; Jennifer J. Quinn (2002). "A Stirling Encounter with Harmonic Numbers" (PDF).
*Mathematics Magazine*.**75**(2): 95–103. doi:10.2307/3219141. - Donald Knuth (1997). "Section 1.2.7: Harmonic Numbers".
*The Art of Computer Programming*. Volume 1:*Fundamental Algorithms*(Third ed.). Addison-Wesley. pp. 75–79. ISBN 0-201-89683-4. - Ed Sandifer,
*How Euler Did It — Estimating the Basel problem*(2003) - Paule, Peter; Schneider, Carsten (2003). "Computer Proofs of a New Family of Harmonic Number Identities" (PDF).
*Adv. Appl. Math*.**31**(2): 359–378. doi:10.1016/s0196-8858(03)00016-2. - Wenchang Chu (2004). "A Binomial Coefficient Identity Associated with Beukers' Conjecture on Apery Numbers" (PDF).
*The Electronic Journal of Combinatorics*.**11**: N15. - Ayhan Dil; István Mező (2008). "A Symmetric Algorithm for Hyperharmonic and Fibonacci Numbers".
*Applied Mathematics and Computation*.**206**(2): 942–951. arXiv:0803.4388 . doi:10.1016/j.amc.2008.10.013.

## External links[edit]

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