Harmonic number explained

In mathematics, the -th harmonic number is the sum of the reciprocals of the first natural numbers:^[1] $$H\_n=\; 1+\backslash frac+\backslash frac+\backslash cdots+\backslash frac\; =\backslash sum\_^n\; \backslash frac.$$

Starting from, the sequence of harmonic numbers begins:$$1,\; \backslash frac,\; \backslash frac,\; \backslash frac,\; \backslash frac,\; \backslash dots$$

Harmonic numbers are related to the harmonic mean in that the -th harmonic number is also times the reciprocal of the harmonic mean of the first 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 most-valuable items is proportional to the -th harmonic number. This leads to a variety of surprising conclusions regarding the long tail and the theory of network value.

The Bertrand-Chebyshev theorem implies that, except for the case, the harmonic numbers are never integers.^[2]

n !! colspan="4"

Harmonic number, Hn
expressed as a fraction		decimal	relative size
1	1
2	3	/2
3	11	/6	~
4	25	/12	~
5	137	/60	~
6	49	/20
7	363	/140	~
8	761	/280	~
9	7 129	/2 520	~
10	7 381	/2 520	~
11	83 711	/27 720	~
12	86 021	/27 720	~
13	1 145 993	/360 360	~
14	1 171 733	/360 360	~
15	1 195 757	/360 360	~
16	2 436 559	/720 720	~
17	42 142 223	/12 252 240	~
18	14 274 301	/4 084 080	~
19	275 295 799	/77 597 520	~
20	55 835 135	/15 519 504	~
21	18 858 053	/5 173 168	~
22	19 093 197	/5 173 168	~
23	444 316 699	/118 982 864	~
24	1 347 822 955	/356 948 592	~
25	34 052 522 467	/8 923 714 800	~
26	34 395 742 267	/8 923 714 800	~
27	312 536 252 003	/80 313 433 200	~
28	315 404 588 903	/80 313 433 200	~
29	9 227 046 511 387	/2 329 089 562 800	~
30	9 304 682 830 147	/2 329 089 562 800	~
31	290 774 257 297 357	/72 201 776 446 800	~
32	586 061 125 622 639	/144 403 552 893 600	~
33	53 676 090 078 349	/13 127 595 717 600	~
34	54 062 195 834 749	/13 127 595 717 600	~
35	54 437 269 998 109	/13 127 595 717 600	~
36	54 801 925 434 709	/13 127 595 717 600	~
37	2 040 798 836 801 833	/485 721 041 551 200	~
38	2 053 580 969 474 233	/485 721 041 551 200	~
39	2 066 035 355 155 033	/485 721 041 551 200	~
40	2 078 178 381 193 813	/485 721 041 551 200	~

Identities involving harmonic numbers

By definition, the harmonic numbers satisfy the recurrence relation$$H\_\; =\; H\_\; +\; \backslash frac.$$

The harmonic numbers are connected to the Stirling numbers of the first kind by the relation$$H\_n\; =\; \backslash frac\backslash left[\{n+1\; \backslash atop\; 2\}\backslash right].$$

The harmonic numbers satisfy the series identities$$\backslash sum\_^n\; H\_k\; =\; (n+1)\; H\_\; -\; n$$and$$\backslash sum\_^n\; H\_k^2\; =\; (n+1)H\_^2\; -\; (2\; n\; +1)\; H\_n\; +\; 2\; n.$$These two results are closely analogous to the corresponding integral results$$\backslash int\_0^x\; \backslash log\; y\; \backslash \; d\; y\; =\; x\; \backslash log\; x\; -\; x$$and$$\backslash int\_0^x\; (\backslash log\; y)^2\backslash \; d\; y\; =\; x\; (\backslash log\; x)^2\; -\; 2\; x\; \backslash log\; x\; +\; 2\; x.$$

Identities involving

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

Calculation

An integral representation given by Euler^[4] is$$H\_n\; =\; \backslash int\_0^1\; \backslash frac\backslash ,dx.$$

The equality above is straightforward by the simple algebraic identity$$\backslash frac=1+x+\backslash cdots\; +x^.$$

Using the substitution, another expression for is

The th harmonic number is about as large as the natural logarithm of . The reason is that the sum is approximated by the integral$$\backslash int\_1^n\; \backslash frac\backslash ,\; dx,$$whose value is .

The values of the sequence decrease monotonically towards the limit$$\backslash lim\_\; \backslash left(H\_n\; -\; \backslash ln\; n\backslash right)\; =\; \backslash gamma,$$where is the Euler–Mascheroni constant. The corresponding asymptotic expansion iswhere are the Bernoulli numbers.

Generating functions

A generating function for the harmonic numbers is$$\backslash sum\_^\backslash infty\; z^n\; H\_n\; =\; \backslash frac,$$where ln(z) is the natural logarithm. An exponential generating function is$$\backslash sum\_^\backslash infty\; \backslash frac\; H\_n\; =\; e^z\; \backslash sum\_^\backslash infty\; \backslash frac\; \backslash frac\; =\; e^z\; \backslash operatorname(z)$$where Ein(z) is the entire exponential integral. The exponential integral may also be expressed as$$\backslash operatorname(z)\; =\; \backslash mathrm\_1(z)\; +\; \backslash gamma\; +\; \backslash ln\; z\; =\; \backslash Gamma\; (0,z)\; +\; \backslash gamma\; +\; \backslash ln\; z$$where Γ(0, z) is the incomplete gamma function.

Arithmetic properties

The harmonic numbers have several interesting arithmetic properties. It is well-known that $H\_n$ is an integer if and only if $n=1$, a result often attributed to Taeisinger.^[5] Indeed, using 2-adic valuation, it is not difficult to prove that for $n\; \backslash ge\; 2$ the numerator of $H\_n$ is an odd number while the denominator of $H\_n$ is an even number. More precisely,$$H\_n=\backslash frac\backslash frac$$with some odd integers $a\_n$ and $b\_n$.

As a consequence of Wolstenholme's theorem, for any prime number

the numerator of is divisible by $p^2$. Furthermore, Eisenstein^[6] proved that for all odd prime number $p$ it holds$$H\_\; \backslash equiv\; -2q\_p(2)\; \backslash pmod\; p$$where $q\_p(2)\; =\; (2^\; -1)/p$ is a Fermat quotient, with the consequence that $p$ divides the numerator of if and only if $p$ 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$$\backslash \backslash subseteq\; J\_p$$for all prime numbers and they defined harmonic primes to be the primes $p$ 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^[8] verified that is finite for all prime numbers up to except 83, 127, and 397; and he gave a heuristic suggesting that the 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

The harmonic numbers appear in several calculation formulas, such as the digamma function$$\backslash psi(n)\; =\; H\_\; -\; \backslash gamma.$$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 using the limit introduced earlier:$$\backslash gamma\; =\; \backslash lim\_,$$although$$\backslash gamma\; =\; \backslash lim\_$$converges more quickly.

In 2002, Jeffrey Lagarias proved^[11] that the Riemann hypothesis is equivalent to the statement that$$\backslash sigma(n)\; \backslash le\; H\_n\; +\; (\backslash log\; H\_n)e^,$$is true for every integer with strict inequality if ; here denotes the sum of the divisors of .

The eigenvalues of the nonlocal problem on

$$\backslash lambda\; \backslash varphi(x)\; =\; \backslash int\_^\; \backslash frac$$ \, dy are given by , where by convention , and the corresponding eigenfunctions are given by the Legendre polynomials .^[12]

Generalizations

Generalized harmonic numbers

The nth generalized harmonic number of order m is given by$$H\_=\backslash sum\_^n\; \backslash frac.$$

(In some sources, this may also be denoted by $H\_n^$ or $H\_m(n).$)

The special case m = 0 gives

The special case m = 1 reduces to the usual harmonic number:$$H\_\; =\; H\_n\; =\; \backslash sum\_^n\; \backslash frac.$$

The limit of $H\_$ as is finite if, with the generalized harmonic number bounded by and converging to the Riemann zeta function$$\backslash lim\_\; H\_\; =\; \backslash zeta(m).$$

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, ...

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$$\backslash int\_0^a\; H\_\; \backslash ,\; dx\; =\; a\; \backslash frac\; -H\_$$and$$\backslash int\_0^a\; H\_\; \backslash ,\; dx\; =\; a\; A\; -\; \backslash frac\; H\_,$$ where A is Apéry's constant ζ(3),and$$\backslash sum\_^n\; H\_=(n+1)H\_-\; H\_\; \backslash text\; m\; \backslash geq\; 0\; .$$

Every generalized harmonic number of order m can be written as a function of harmonic numbers of order

using$$H\_\; =\; \backslash sum\_^\; \backslash frac\; +\; \backslash frac$$   for example:

A generating function for the generalized harmonic numbers is$$\backslash sum\_^\backslash infty\; z^n\; H\_\; =\; \backslash frac,$$where

is the polylogarithm, and . The generating function given above for 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:$$H\_=\backslash zeta(m)-p^m\backslash sum\_^\backslash infty\; \backslash frac$$where is the Riemann zeta function. The relevant recurrence relation is$$H\_=H\_+\backslash frac.$$Some special values arewhere G is Catalan's constant. In the special case that , we get$$H\_=\backslash zeta(m,\; 1)\; -\; \backslash zeta(m,\; n+1),$$

where

is the Hurwitz zeta function. This relationship is used to calculate harmonic numbers numerically.

Multiplication formulas

The multiplication theorem applies to harmonic numbers. Using polygamma functions, we obtainor, more generally,$$H\_=\backslash frac\backslash left(H\_x+H\_+H\_+\backslash cdots\; +H\_\; \backslash right)\; +\; \backslash ln\; n.$$

For generalized harmonic numbers, we havewhere

is the Riemann zeta function.

Hyperharmonic numbers

The next generalization was discussed by J. H. Conway and R. K. Guy in their 1995 book The Book of Numbers. Let$$H\_n^\; =\; \backslash frac1n.$$Then the nth hyperharmonic number of order r (r>0) is defined recursively as$$H\_n^\; =\; \backslash sum\_^n\; H\_k^.$$In particular,

is the ordinary harmonic number .

Roman Harmonic numbers

The Roman Harmonic numbers,^[13] named after Steven Roman, were introduced by Daniel Loeb and Gian-Carlo Rota in the context of a generalization of umbral calculus with logarithms.^[14] There are many possible definitions, but one of them, for

, is$$c\_n^\; =\; 1,$$and$$c\_n^\; =\; \backslash sum\_^n\backslash frac.$$Of course,$$c\_n^\; =\; H\_n.$$

If

, they satisfy$$c\_n^\; -\; \backslash frac\; =\; c\_^.$$Closed form formulas are$$c\_n^\; =\; n!\; (-1)^k\; s(-n,k),$$where is Stirling numbers of the first kind generalized to negative first argument, and$$c\_n^\; =\; \backslash sum\_^n\; \backslash binom\; \backslash frac,$$which was found by Donald Knuth.

In fact, these numbers were defined in a more general manner using Roman numbers and Roman factorials, that include negative values for

. This generalization was useful in their study to define Harmonic logarithms.

Harmonic numbers for real and complex values

The formulae given above,$$H\_x\; =\; \backslash int\_0^1\; \backslash frac\; \backslash ,\; dt=\; \backslash sum\_^\backslash infty\; \backslash frac$$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$$H\_x\; =\; \backslash psi(x+1)+\backslash gamma,$$where is the digamma function, and is the Euler–Mascheroni constant. The integration process may be repeated to obtain$$H\_=\; \backslash sum\_^\backslash infty\; \backslash frac\; H\_k.$$

The Taylor series for the harmonic numbers iswhich comes from the Taylor series for the digamma function (

is the Riemann zeta function).

Alternative, asymptotic formulation

When seeking to approximate  for a complex number , it is effective to first compute  for some large integer . Use that as an approximation for the value of . Then use the recursion relation backwards  times, to unwind it to an approximation for . Furthermore, this approximation is exact in the limit as  goes to infinity.

Specifically, for a fixed integer , it is the case that$$\backslash lim\_\; \backslash left[H\_\{m+n\}\; -\; H\_m\backslash right]\; =\; 0.$$

If  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  is replaced by an arbitrary complex number ,

$$\backslash lim\_\; \backslash left[H\_\{m+x\}\; -\; H\_m\backslash right]\; =\; 0\backslash ,.$$Swapping the order of the two sides of this equation and then subtracting them from  gives

This infinite series converges for all complex numbers  except the negative integers, which fail because trying to use the recursion relation backwards through the value  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), (2) for all complex numbers  except the non-positive integers, and (3) for all complex values .

This last formula can be used to show that$$\backslash int\_0^1\; H\_x\; \backslash ,\; dx\; =\; \backslash gamma,$$where  is the Euler–Mascheroni constant or, more generally, for every  we have:$$\backslash int\_0^nH\_\backslash ,dx\; =\; n\backslash gamma\; +\; \backslash ln(n!)\; .$$

Special values for fractional arguments

There are the following special analytic values for fractional arguments between 0 and 1, given by the integral$$H\_\backslash alpha\; =\; \backslash int\_0^1\backslash frac\backslash ,dx\backslash ,\; .$$

More values may be generated from the recurrence relation$$H\_\backslash alpha\; =\; H\_+\backslash frac\backslash ,,$$or from the reflection relation$$H\_-H\_\backslash alpha\; =\; \backslash pi\backslash cot-\backslash frac+\backslash frac\backslash ,\; .$$

For example:

Which are computed via Gauss's digamma theorem, which essentially states that for positive integers p and q with p < q$$H\_\; =\; \backslash frac\; +2\backslash sum\_^\; \backslash cos\backslash left(\backslash frac\backslash right)\backslash ln\backslash left(\backslash right)-\backslash frac\backslash cot\backslash left(\backslash frac\backslash right)-\backslash ln\backslash left(2q\backslash right)$$

Relation to the Riemann zeta function

Some derivatives of fractional harmonic numbers are given by

And using Maclaurin series, we have for x < 1 that

For fractional arguments between 0 and 1 and for a > 1,

See also

• Watterson estimator
• Tajima's D
• Coupon collector's problem
• Jeep problem
• 100 prisoners problem
• Riemann zeta function
• List of sums of reciprocals
• False discovery rate#Benjamini–Yekutieli procedure
• Block-stacking problem

References

• Arthur T. Benjamin . Gregory O. Preston . Jennifer J. Quinn . A Stirling Encounter with Harmonic Numbers . 2002 . . 75 . 2 . 95–103 . 10.2307/3219141 . 3219141 . 10.1.1.383.722 . 2005-08-08 . https://web.archive.org/web/20090617002133/http://www.math.hmc.edu/~benjamin/papers/harmonic.pdf . 2009-06-17 . dead .
• Book: Donald Knuth . Donald Knuth . The Art of Computer Programming . 1: Fundamental Algorithms . Third . Addison-Wesley . 1997 . 978-0-201-89683-1 . Section 1.2.7: Harmonic Numbers . 75–79.
• Ed Sandifer, How Euler Did It — Estimating the Basel problem (2003)
• Paule . Peter . Peter Paule . Schneider . Carsten . 10.1016/s0196-8858(03)00016-2 . 2 . Adv. Appl. Math. . 359–378 . Computer Proofs of a New Family of Harmonic Number Identities . 31 . 2003.
• Wenchang Chu . A Binomial Coefficient Identity Associated with Beukers' Conjecture on Apery Numbers . 2004 . The Electronic Journal of Combinatorics . 11 . N15. 10.37236/1856 . free .
• Ayhan Dil . István Mező . A Symmetric Algorithm for Hyperharmonic and Fibonacci Numbers . 2008 . Applied Mathematics and Computation . 206 . 2 . 942–951 . 10.1016/j.amc.2008.10.013. 0803.4388 . 12130670 .

Notes and References

1. Book: Knuth, Donald . The Art of Computer Programming . Addison-Wesley . 1997 . 0-201-89683-4 . 3rd . 75-79 . en.
2. Book: Ronald L. . Graham. Donald E. . Knuth. Oren . Patashnik. Concrete Mathematics. 1994. Addison-Wesley. Concrete Mathematics.
3. Web site: Weisstein . Eric W. . Harmonic Number . 2024-09-30 . mathworld.wolfram.com . en.
4. .
5. Book: Weisstein, Eric W.. CRC Concise Encyclopedia of Mathematics. Chapman & Hall/CRC. 2003. 978-1-58488-347-0. Boca Raton, FL. 3115.
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. 3. 249–257. 10.1016/0012-365X(90)90234-9. free.
8. Boyd. David W.. 1994. A p-adic study of the partial sums of the harmonic series. Experimental Mathematics. 3. 4. 287–302. 10.1080/10586458.1994.10504298. 10.1.1.56.7026.
9. Sanna. Carlo. 2016. On the p-adic valuation of harmonic numbers. Journal of Number Theory. 166. 41–46. 10.1016/j.jnt.2016.02.020. 2318/1622121. free.
10. Chen. Yong-Gao. Wu. Bing-Ling. 2017. On certain properties of harmonic numbers. Journal of Number Theory. 175. 66–86. 10.1016/j.jnt.2016.11.027.
11. Jeffrey Lagarias . An Elementary Problem Equivalent to the Riemann Hypothesis . Amer. Math. Monthly . 109 . 6 . 2002 . 534–543 . math.NT/0008177 . 10.2307/2695443. 2695443 .
12. E.O. Tuck . Some methods for flows past blunt slender bodies . J. Fluid Mech. . 18 . 1964 . 4 . 619–635 . 10.1017/S0022112064000453. 1964JFM....18..619T . 123120978 .
13. Sesma . J. . 2017 . The Roman harmonic numbers revisited . Journal of Number Theory . 180 . 544–565 . 10.1016/j.jnt.2017.05.009 . 0022-314X. 1702.03718 .
14. Loeb . Daniel E . Rota . Gian-Carlo . 1989 . Formal power series of logarithmic type . Advances in Mathematics . 75 . 1 . 1–118 . 10.1016/0001-8708(89)90079-0 . 0001-8708. free .