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: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.
|n||Harmonic number, Hn|
|expressed as a fraction||decimal||relative size|
|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|
Identities involving harmonic numbers
By definition, the harmonic numbers satisfy the recurrence relation
The harmonic numbers are connected to the Stirling numbers of the first kind by the relation
satisfy the property
is an integral of the logarithmic function.
The harmonic numbers satisfy the series identity
An integral representation given by Euler is
The equality above is straightforward by the simple algebraic identity
Using the substitution x = 1 − u, another expression for Hn is
A closed form expression for Hn is
whose value is ln n.
The values of the sequence Hn − ln n decrease monotonically towards the limit
where Bk are the Bernoulli numbers.
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.
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. 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 proved that for all odd prime number it holds
for all prime numbers and they defined harmonic primes to be the primes such that has exactly 3 elements.
Eswarathasan and Levine also conjectured that is a finite set for all primes and that there are infinitely many harmonic primes. Boyd verified that is finite for all prime numbers up to except 83, 127, and 397; and he gave an heuristic suggesting that the density of the harmonic primes in the set of all primes should be . Sanna showed that has zero asymptotic density, while Bing-Ling Wu and Yong-Gao Chen proved that the number of elements of not exceeding is at most , for all .
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:
converges more quickly.
The eigenvalues of the nonlocal problem
are given by , where by convention,
Generalized harmonic numbers
The generalized harmonic number of order m of n is given by
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 limit as n → ∞ is finite if m > 1, with the generalized harmonic number bounded by and converging to the Riemann zeta function
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)
Some integrals of generalized harmonic numbers are
- where A is Apéry's constant, i.e. ζ(3).
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
In the special case that , we get
- where is the Hurwitz zeta function. This relationship is used to calculate harmonic numbers numerically.
or, more generally,
For generalized harmonic numbers, we have
where is the Riemann zeta function.
Harmonic numbers for real and complex values
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
When seeking to approximate Hx for a complex number x, it is effective to first compute Hm for some large integer m. 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 a fixed integer n, it is the case that
If n is not an integer then it is not possible to say whether this equation is true because we have not yet (in this section) defined harmonic numbers for non-integers. However, we do get a unique extension of the harmonic numbers to the non-integers by insisting that this equation continue to hold when the arbitrary integer n is replaced by an arbitrary complex number x.
Swapping the order of the two sides of this equation and then subtracting them from Hx gives
This infinite series converges for all complex numbers 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 this construction, the function that defines the harmonic number for complex values is the unique function that simultaneously satisfies (1) H0 = 0, (2) Hx = Hx−1 + 1/x for all complex numbers x except the non-positive integers, and (3) limm→+∞ (Hm+x − Hm) = 0 for all complex values x.
Note that this last formula can be used to show that:
where γ is the Euler–Mascheroni constant or, more generally, for every n we have:
Special values for fractional arguments
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 positive integers p and q with p < q, we have:
Relation to the Riemann zeta function
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:
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