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]

The first 40 harmonic numbers
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[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


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 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.

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 .

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[9] 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 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

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 → +∞ 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.

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.

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 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 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]

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. ^ 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. ^ Jeffrey Lagarias (2002). "An Elementary Problem Equivalent to the Riemann Hypothesis". Amer. Math. Monthly. 109: 534–543. arXiv:math.NT/0008177Freely accessible. doi:10.2307/2695443. 

References[edit]

External links[edit]

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