Harmonic number
For other uses, see Harmonic number (disambiguation).
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 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 implies that, except for the case n = 1, the harmonic numbers are never integers.
| n | Harmonic number, Hn | |||
|---|---|---|---|---|
| expressed as a fraction | decimal | relative size | ||
| 1 | 1 | 1 | 1 | |
| 2 | 3 | /2 | 1.5 | 1.5 |
| 3 | 11 | /6 | ~1.83333 | 1.83333 |
| 4 | 25 | /12 | ~2.08333 | 2.08333 |
| 5 | 137 | /60 | ~2.28333 | 2.28333 |
| 6 | 49 | /20 | 2.45 | 2.45 |
| 7 | 363 | /140 | ~2.59286 | 2.59286 |
| 8 | 761 | /280 | ~2.71786 | 2.71786 |
| 9 | 7 129 | /2 520 | ~2.82897 | 2.82897 |
| 10 | 7 381 | /2 520 | ~2.92897 | 2.92897 |
| 11 | 83 711 | /27 720 | ~3.01988 | 3.01988 |
| 12 | 86 021 | /27 720 | ~3.10321 | 3.10321 |
| 13 | 1 145 993 | /360 360 | ~3.18013 | 3.18013 |
| 14 | 1 171 733 | /360 360 | ~3.25156 | 3.25156 |
| 15 | 1 195 757 | /360 360 | ~3.31823 | 3.31823 |
| 16 | 2 436 559 | /720 720 | ~3.38073 | 3.38073 |
| 17 | 42 142 223 | /12 252 240 | ~3.43955 | 3.43955 |
| 18 | 14 274 301 | /4 084 080 | ~3.49511 | 3.49511 |
| 19 | 275 295 799 | /77 597 520 | ~3.54774 | 3.54774 |
| 20 | 55 835 135 | /15 519 504 | ~3.59774 | 3.59774 |
| 21 | 18 858 053 | /5 173 168 | ~3.64536 | 3.64536 |
| 22 | 19 093 197 | /5 173 168 | ~3.69081 | 3.69081 |
| 23 | 444 316 699 | /118 982 864 | ~3.73429 | 3.73429 |
| 24 | 1 347 822 955 | /356 948 592 | ~3.77596 | 3.77596 |
| 25 | 34 052 522 467 | /8 923 714 800 | ~3.81596 | 3.81596 |
| 26 | 34 395 742 267 | /8 923 714 800 | ~3.85442 | 3.85442 |
| 27 | 312 536 252 003 | /80 313 433 200 | ~3.89146 | 3.89146 |
| 28 | 315 404 588 903 | /80 313 433 200 | ~3.92717 | 3.92717 |
| 29 | 9 227 046 511 387 | /2 329 089 562 800 | ~3.96165 | 3.96165 |
| 30 | 9 304 682 830 147 | /2 329 089 562 800 | ~3.99499 | 3.99499 |
| 31 | 290 774 257 297 357 | /72 201 776 446 800 | ~4.02725 | 4.02725 |
| 32 | 586 061 125 622 639 | /144 403 552 893 600 | ~4.05850 | 4.0585 |
| 33 | 53 676 090 078 349 | /13 127 595 717 600 | ~4.08880 | 4.0888 |
| 34 | 54 062 195 834 749 | /13 127 595 717 600 | ~4.11821 | 4.11821 |
| 35 | 54 437 269 998 109 | /13 127 595 717 600 | ~4.14678 | 4.14678 |
| 36 | 54 801 925 434 709 | /13 127 595 717 600 | ~4.17456 | 4.17456 |
| 37 | 2 040 798 836 801 833 | /485 721 041 551 200 | ~4.20159 | 4.20159 |
| 38 | 2 053 580 969 474 233 | /485 721 041 551 200 | ~4.22790 | 4.2279 |
| 39 | 2 066 035 355 155 033 | /485 721 041 551 200 | ~4.25354 | 4.25354 |
| 40 | 2 078 178 381 193 813 | /485 721 041 551 200 | ~4.27854 | 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
The functions
satisfy the property
In particular
is an integral of the logarithmic function.
The harmonic numbers satisfy the series identities
these two results are closely analogous to the corresponding integral results
Identities involving π
There are several infinite summations involving harmonic numbers and powers of π:
Calculation
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
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 is
where Bk are the Bernoulli numbers.
Generating functions
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
Applications
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 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
Generalizations
Generalized harmonic numbers
The generalized harmonic number of order m of n is given by
Other notations occasionally used include
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 k 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 in the OEIS)
Some integrals of generalized harmonic numbers are
and
and
Every generalized harmonic number of order m can be written as a function of harmonic of order m-1 using:
A generating function for the generalized harmonic numbers is
A fractional argument for generalized harmonic numbers can be introduced as follows:
Some special values are:
Multiplication formulas
The multiplication theorem applies to harmonic numbers.
Using polygamma functions, we obtain
or, more generally,
For generalized harmonic numbers, we have
Hyperharmonic numbers
The next generalization was discussed by J. H. Conway and R. K. Guy in their 1995 book The Book of Numbers.
Let
Then the nth hyperharmonic number of order r (r>0) is defined recursively as
Harmonic numbers for real and complex values
See also
- Watterson estimator
- Tajima's D
- Coupon collector's problem
- Jeep problem
- Riemann zeta function
- List of sums of reciprocals
Credits to the contents of this page go to the authors of the corresponding Wikipedia page: en.wikipedia.org/wiki/Harmonic number.