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Harmonic Number

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HarmonicNumber

A harmonic number is a number of the form

H_n=sum_(k=1)^n1/k
(1)

for a positive integer n, arising from truncation of the harmonic series, with H_0=0. A harmonic number can be expressed analytically as

H_n=gamma+psi_0(n+1),
(2)

where gamma is the Euler-Mascheroni constant and Psi(x)=psi_0(x) is the digamma function.

The first few harmonic numbers H_n are 1, 3/2, 11/6, 25/12, 137/60, ... (OEIS A001008 and A002805). The numbers of digits in the numerator of H_(10^n) for n=0, 1, ... are 1, 4, 41, 434, 4346, 43451, 434111, 4342303, 43428680, ... (OEIS A114467), with the corresponding number of digits in the denominator given by 1, 4, 40, 433, 4345, 43450, 434110, 4342302, 43428678, ... (OEIS A114468). The ratios of these digit counts to 10^n appear to approach log_(10)e=0.43429448... (OEIS A002285).

HarmonicNumberPrimes

The first few indices n such that the numerator of H_n is prime are given by 2, 3, 5, 8, 9, 21, 26, 41, 56, 62, 69, ... (OEIS A056903). E. W. Weisstein reported completing the search for prime numerators up to 81780 on May 13, 2009. The following table summarizes the largest values known at that time.

ndecimal digitsdiscoverer
6394227795E. W. Weisstein (Feb. 14, 2007)
6929430067E. W. Weisstein (Feb. 1, 2008)
6992730301E. W. Weisstein (Mar. 11, 2008)
7744933616E. W. Weisstein (Apr. 4, 2009)
7812833928E. W. Weisstein (Apr. 9, 2009)
7899334296E. W. Weisstein (Apr. 17, 2009)
8165835479E. W. Weisstein (May. 12, 2009)

The denominators of H_n appear never to be prime except for the case H_2=3/2. Furthermore, the denominator is never a prime power (except for this case) since the denominator is always divisible by the largest power of 2 less than or equal to n, and also by any prime p with n/2<p<=n.

The harmonic numbers are implemented as HarmonicNumber[n].

The values of n such that H_n equals or exceeds 1, 2, 3, ... are given by 1, 4, 11, 31, 83, 227, 616, 1674, ... (OEIS A004080). Another interesting sequence is the number of terms in the simple continued fraction of H_(10^n) for n=0, 1, 2, ..., given by 1, 8, 68, 834, 8356, 84548, 841817, 8425934, 84277586, ... (OEIS A091590). The length divided by 10^n is conjectured to approach 12ln2/pi^2=0.8427659... (OEIS A089729).

HarmonicNumberReIm
HarmonicNumberContours

The definition of harmonic numbers can also be extended to the complex plane, as illustrated above.

Based on their definition, harmonic numbers satisfy the obvious recurrence equation

H_n=1/n+H_(n-1)
(3)

with H_1=1.

The related alternating harmonic numbers are obtained by alternating the signs in the defining sum.

For n>1, the harmonic number H_n is never an integer, which can be proved by using the strong triangle inequality to show that the p-adic norm of H_n with p=2 is greater than 1 for n>1. This result was proved in 1915 by Taeisinger, and the more general result that any sum of two or more consecutive reciprocals of positive integers, not necessarily starting with 1, is never an integer was proved by Kűrschák in 1918 (Hoffman 1998, p. 157).

For n>1, the harmonic numbers have odd numerators and even denominators. The nth harmonic number is given asymptotically by

H_n∼lnn+gamma+1/(2n)-1/(12)n^(-2)+1/(120)n^(-4)-1/(252)n^(-6)+...,
(4)

where gamma is the Euler-Mascheroni constant (Conway and Guy 1996; Havil 2003, pp. 79 and 89). The coefficient of n^(-2k) is zeta(1-2k), where zeta(s) is the Riemann zeta function. These coefficients are -1/12, 1/120, -1/252, 1/240, ... for k=1, 2, .... Their reduced positive denominators are 12, 120, 252, 240, ... (OEIS A006953). This formula is a special case of the Euler-Maclaurin integration formulas (Havil 2003, p. 79).

B. Cloitre (pers. comm., Jan. 1, 2021) gave the asymptotic formula for the product

product_(k=1)^n(n+k+H_k)/(n+k)∼2^gammae^(-pi^2/12)n^(ln2)
(5)

as the positive integer n->infty, where gamma is the Euler-Mascheroni constant.

B. Cloitre (pers. comm., Oct. 29, 2026) also gave the generalization

product_(k=1)^n(1+(rH_k)/(n+k))∼2^(rgamma)e^(-rpi^2/12)n^(rln2)
(6)

for any fixed real number r as the positive integer n->infty, together with the product limit

lim_(n->infty)product_(k=1)^(n-1)(1+((H_n-H_k)(H_n-H_(n-k)))/k)=e^(zeta(3)),
(7)

where zeta(3) is Apéry's constant.

HarmonicNumberInequalities

Inequalities bounding H_n include

1/(2(n+1))<H_n-lnn-gamma<1/(2n)
(8)

(Young 1991; Havil 2003, pp. 73-75) and

1/(24(n+1)^2)<H_n-ln(n+1/2)-gamma<1/(24n^2)
(9)

(DeTemple 1991; Havil 2003, pp. 76-78).

An interesting analytic sum is given by

sum_(n=1)^infty(H_n)/(n·2^n)=1/(12)pi^2.
(10)

(Coffman 1987). Borwein and Borwein (1995) show that

sum_(n=1)^(infty)(H_n^2)/((n+1)^2)=(11)/4zeta(4)=(11)/(360)pi^4
(11)
sum_(n=1)^(infty)(H_n^2)/(n^2)=(17)/4zeta(4)=(17)/(360)pi^4
(12)
sum_(n=1)^(infty)(H_n)/(n^3)=5/4zeta(4)=1/(72)pi^4
(13)
sum_(n=1)^(infty)(H_n)/(n^4)=3zeta(5)-1/6pi^2zeta(3)
(14)
sum_(n=1)^(infty)(H_n)/(n^5)=1/(540)pi^6-1/2[zeta(3)]^2,
(15)

where zeta(z) is the Riemann zeta function. The first of these had been previously derived by de Doelder (1991), and the third by Goldbach in a 1742 letter to Euler (Borwein and Bailey 2003, pp. 99-100; Bailey et al. 2007, p. 256). These identities are corollaries of the identity

1/piint_0^pix^2{ln[2cos(1/2x)]}^2dx=(11)/2zeta(4)=(11)/(180)pi^4
(16)

(Borwein and Borwein 1995). Additional identities due to Euler are

sum_(n=1)^(infty)(H_n)/(n^2)=2zeta(3)
(17)
2sum_(n=1)^(infty)(H_n)/(n^m)=(m+2)zeta(m+1)-sum_(n=1)^(m-2)zeta(m-n)zeta(n+1)
(18)

for m=2, 3, ... (Borwein and Borwein 1995), where zeta(3) is Apéry's constant. These sums are related to so-called Euler sums.

M. Bindhammer (pers. comm., Jul. 4, 2022) communicated the conditionally convergent identity

sum_(n=1)^infty((-1)^(n-1)(2H_(2n)-H_n))/n=1/8pi^2.
(19)

A general identity due to B. Cloitre (pers. comm., Jan. 7, 2006) is

sum_(k=1)^infty(H_k)/((k+1)_m)=1/((m-1)!(m-1)^2)
(20)

for integers m>=2, where (x)_n is a Pochhammer symbol.

Gosper (1996) gave the interesting identity

sum_(i=0)^(infty)(z^iH_i)/(i!)=-e^zsum_(k=1)^(infty)((-z)^k)/(kk!)
(21)
=e^z[lnz+Gamma(0,z)+gamma],
(22)

where Gamma(0,z) is the incomplete gamma function and gamma is the Euler-Mascheroni constant.

G. Huvent (2002) found the beautiful formula

zeta(5)=-(16)/(11)sum_(n=1)^infty([2(-1)^n+1]H_n)/(n^4).
(23)

A beautiful double series is given by

sum_(k=1)^inftysum_(j=1)^infty(H_j(H_(k+1)-1))/(kj(k+1)(j+k)) =-4zeta(2)-2zeta(3)+4zeta(2)zeta(3)+2zeta(5)
(24)

(Bailey et al. 2007, pp. 273-274). Another double sum is

sum_(i=0)^(k-1)sum_(j=k)^n((-1)^(i+j-1))/(j-i)(n; i)(n; j)=sum_(i=0)^(k-1)(n; i)^2(H_(n-i)-H_i)
(25)

for 1<=k<=n (Sondow 2003, 2005).

There is an unexpected connection between the harmonic numbers and the Riemann hypothesis.

Generalized harmonic numbers in power r, for nonnegative integers n, can be defined by the relationship

H_(n,r)=sum_(k=1)^n1/(k^r),
(26)

where H_(n,1)=H_n and H_(0,r)=0. These numbers are implemented as HarmonicNumber[n, r].

The numerators of the special case H_(n,2) are known as Wolstenholme numbers. B. Cloitre (pers. comm., ) gave the surprising identity

H_(n,2)=1/2sum_(i=1)^nsum_(j=1)^n((i-1)!(j-1)!)/((i+j)!)+3/2sum_(k=1)^n1/(k^2(2k; k))
(27)

which relates H_(n,2) to an indefinite version of a famous series for zeta(2). H_(n,2) also satisfies

lim_(n->infty)H_(n,2)=zeta(2)=(pi^2)/6,
(28)

where zeta(2) is the Riemann zeta function. This follows from the identity

H_(n,2)=zeta(2)-psi_1(n+1),
(29)

where psi_1(z) is the trigamma function since

lim_(n->infty)psi_1(n+1)=0.
(30)

For n>=1 and odd r>=3, the generalized harmonic numbers have the explicit form

H_(n,r)=n^(-r)+(psi_(r-1)(n))/(Gamma(r))+zeta(r),
(31)

where psi_r(n) is the polygamma function, Gamma(r) is the gamma function, and zeta(r) is the Riemann zeta function.

The 2-index harmonic numbers satisfy the identity

H_(n,r)=2^(r-1)(H_(2n,r)-H_(2n,r)^')
(32)

(P. Simon, pers. comm., Aug. 30, 2004), where H_(n,r)^'=sum_(k=1)^(n)(-1)^(k+1)/k^r.

Sums of the generalized harmonic numbers H_(n,r) include

sum_(n=1)^inftyH_(n,r)z^n=(Li_r(z))/(1-z)
(33)

for |z|<1, where Li_r(z) is a polylogarithm,

sum_(k=1)^(infty)(H_(k,1))/(k(z+1)^k)=-Li_2(-1/z)
(34)
sum_(k=1)^(infty)(H_(k,1))/(kphi^(2k))=1/(15)pi^2-1/2[csch^(-1)2]^2
(35)
sum_(k=1)^(infty)(H_(k,1))/(k^22^k)=zeta(3)-1/(12)pi^2ln2
(36)
sum_(k=1)^(infty)(H_(k,2))/(k^4)=[zeta(3)]^2-(pi^6)/(2835)
(37)
sum_(k=1)^(infty)(H_(k,2))/(k2^k)=5/8zeta(3)
(38)
sum_(k=1)^(infty)(H_(k,4))/(k^2)=(37pi^6)/(11340)-[zeta(3)]^2,
(39)

where Li_2(z) is a dilogarithm and phi is the golden ratio. The first identity requires |z+1|>1. The first, third, fourth, and fifth identities in this display are due to B. Cloitre (pers. comm., Oct. 4, 2004). In general,

sum_(k=1)^infty(H_(k,r))/(k^r)=1/2{[zeta(r)]^2+zeta(2r)}
(40)

for r>1 (P. Simone, pers. comm., June 2, 2003). The power harmonic numbers also obey the unexpected identity

9H_(8,n)-19H_(9,n)+10H_(10,n)+sum_(k=1)^(n-1)[H_(8,n-k)H_(9,k)-H_(9,n-k)H_(9,k) -H_(8,n-k)H_(10,k)+H_(9,n-k)H_(10,k)]=0
(41)

(M. Trott, pers. comm.).

P. Simone (pers. comm., Aug. 30, 2004) showed that

[C(t)]^2+[S(t)]^2=1/(90)pi^4+2/3pi^2C(t) -2sum_(m=1)^infty((H_(m,2))/(m^2)+(2H_m)/(m^3))cos(mt),
(42)

where t is real and

C(t)=sum_(n=1)^(infty)(cos(nt))/(n^2)
(43)
=1/2[Li_2(e^(-it))+Li_2(e^(it))]
(44)
S(t)=sum_(n=1)^(infty)(sin(nt))/(n^2)
(45)
=1/2i[Li_2(e^(-it))-Li_2(e^(it))].
(46)

This gives the special results

sum_(n=1)^infty(H_n)/(n^3)=1/(72)pi^4 1/8sum_(k=1)^infty((2H_(4k,2))/(k^2)+(H_(4k))/(k^3))=(211pi^4)/(11520)-K^2 2sum_(k=1)^infty[((-1)^(k+1)H_(k,2))/(k^2)+(2(-1)^(k+1)H_k)/(k^3)]=(37pi^4)/(720)
(47)

for t=0, pi/2, and pi, respectively, where K is Catalan's constant.

Conway and Guy (1996) define the second-order harmonic number by

H_n^((2))=sum_(i=1)^(n)H_i
(48)
=(n+1)(H_(n+1)-1)
(49)
=(n+1)(H_(n+1)-H_1),
(50)

the third-order harmonic number by

H_n^((3))=sum_(i=1)^nH_i^((2))=(n+2; 2)(H_(n+2)-H_2),
(51)

and the kth-order harmonic number, for positive integer k, by

H_n^((k))=(n+k-1; k-1)(H_(n+k-1)-H_(k-1)).
(52)

A slightly different definition of a two-index harmonic number c_n^((j)) is given by Roman (1992) in connection with the harmonic logarithm. For integer n and nonnegative integer j, Roman (1992) defines this by

c_n^((0))={1 for n>=0; 0 for n<0
(53)
c_0^((j))={1 for j=0; 0 for j!=0
(54)

plus the recurrence relation

nc_n^((j))=c_n^((j-1))+|_n]c_(n-1)^((j))
(55)

for j>0, where |_n] is the Roman symbol. For general n>0 and j>0, this is equivalent to

c_n^((j))=sum_(i=1)^n1/ic_i^((j-1)),
(56)

and for n>0, it simplifies to

c_n^((j))=sum_(i=1)^n(n; i)(-1)^(i-1)i^(-j).
(57)

For n<0, the harmonic number can be written

c_n^((j))=(-1)^j|_n]!s(-n,j),
(58)

where |_n]! is the Roman factorial and s is a Stirling number of the first kind.

A separate type of number sometimes also called a "harmonic number" is a harmonic divisor number (or Ore number).

See also

Alternating Harmonic Number, Apéry's Constant, Book Stacking Problem, Egyptian Fraction, Euler Sum, Faulhaber's Formula, Harmonic Divisor Number, Harmonic Logarithm, Harmonic Series, Roman Factorial, Roman Symbol, Unit Fraction, Wolstenholme Number

Related Wolfram sites

https://functions.wolfram.com/GammaBetaErf/HarmonicNumber/, https://functions.wolfram.com/GammaBetaErf/HarmonicNumber2/

Portions of this entry contributed by Jonathan Sondow (author's link)

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References

Bailey, D. H.; Borwein, J. M.; Calkin, N. J.; Girgensohn, R.; Luke, D. R.; and Moll, V. H. Experimental Mathematics in Action. Wellesley, MA: A K Peters, 2007.Borwein, D. and Borwein, J. M. "On an Intriguing Integral and Some Series Related to zeta(4)." Proc. Amer. Math. Soc. 123, 1191-1198, 1995.Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, 2003.Coffman, S. W. "Problem 1240 and Solution: An Infinite Series with Harmonic Numbers." Math. Mag. 60, pp. 118-119, 1987.Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 143 and 258-259, 1996.de Doelder, P. J. "On Some Series Containing Psi(x)-Psi(y) and (Psi(x)-Psi(y))^2 for Certain Values of x and y." J. Comp. Appl. Math. 37, 125-141, 1991.DeTemple, D. W. "The Non-Integer Property of Sums of Reciprocals of Consecutive Integers." Math. Gaz. 75, 193-194, 1991.Flajolet, P. and Salvy, B. "Euler Sums and Contour Integral Representation." Experim. Math. 7, 15-35, 1998.Gosper, R. W. "harmonic Summation and exponential gfs." math-fun@cs.arizona.edu posting, Aug. 2, 1996.Graham, R. L.; Knuth, D. E.; and Patashnik, O. "Harmonic Numbers" and "Harmonic Summation." §6.3 and 6.4 in Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, pp. 272-282, 1994.Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, 2003.Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth. New York: Hyperion, 1998.Huvent, G. "Autour de la primitive de t^pcoth(alphat/2)." Feb. 3, 2002. http://perso.orange.fr/gery.huvent/articlespdf/Autour_primitive.pdf.Roman, S. The Umbral Calculus. New York: Academic Press, p. 99, 1984.Roman, S. "The Logarithmic Binomial Formula." Amer. Math. Monthly 99, 641-648, 1992.Savio, D. Y.; Lamagna, E. A.; and Liu, S.-M. "Summation of Harmonic Numbers." In Computers and Mathematics (Ed. E. Kaltofen and S. M. Watt). New York: Springer-Verlag, pp. 12-20, 1989.Sloane, N. J. A. Sequences A001008/M2885, A002285/M3210, A002805/M1589, A004080, A006953/M2039, A056903, A082912, A089729, A091590, A096618, A114467, and A114468 in "The On-Line Encyclopedia of Integer Sequences."Sondow, J. "Criteria for Irrationality of Euler's Constant." Proc. Amer. Math. Soc. 131, 3335-3344, 2003.Sondow, J. "Problem 11026: An Identity Involving Harmonic Numbers." Amer. Math. Monthly 112, 367-369, 2005.Trott, M. "The Mathematica Guidebooks Additional Material: Harmonic Numbers Inversion." https://www.mathematicaguidebooks.org/additions.shtml#S_3_06.Young, R. M. "Euler's Constant." Math. Gaz. 75, 187-190, 1991.

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Harmonic Number

Cite this as:

Weisstein, Eric W., with contributions by Jonathan Sondow. "Harmonic Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HarmonicNumber.html

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