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A harmonic number is a number of the form

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

where is the Euler-Mascheroni constant and is the digamma function.

The first few harmonic numbers are 1, , , , , ... (OEIS A001008 and A002805). The numbers of digits in the numerator of for , 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 appear to approach (OEIS A002285).

The first few indices such that the numerator of 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 on May 13, 2009. The following table summarizes the largest indices reported at that time. The larger numerators in this search can include probable primes rather than certified primes.

The denominators of appear never to be prime except for the case . 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 , and also by any prime with .

The harmonic numbers are implemented as HarmonicNumber[n].

The values of such that 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 for , 1, 2, ..., given by 1, 8, 68, 834, 8356, 84548, 841817, 8425934, 84277586, ... (OEIS A091590). The length divided by is conjectured to approach (OEIS A089729).

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

with .

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

For , the harmonic number is never an integer, which can be proved by using the strong triangle inequality to show that the p-adic norm of with is greater than 1 for . 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 , the harmonic numbers have odd numerators and even denominators. The th harmonic number is given asymptotically by

where is the Euler-Mascheroni constant (Conway and Guy 1996; Havil 2003, pp. 79 and 89). The coefficient of is , where is the Riemann zeta function. These coefficients are , , , , ... for , 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

as the positive integer , where is the Euler-Mascheroni constant.

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

for any fixed real number as the positive integer , together with the product limit

where is Apéry's constant.

Inequalities bounding include

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

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

An interesting analytic sum is given by

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

where 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

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

for , 3, ... (Borwein and Borwein 1995), where 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

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

for integers , where is a Pochhammer symbol.

Gosper (1996) gave the interesting identity

where is the incomplete gamma function and is the Euler-Mascheroni constant.

G. Huvent (2002) found the beautiful formula

A beautiful double series is given by

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

for (Sondow 2003, 2005).

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

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

where and . These numbers are implemented as HarmonicNumber[n, r].

Splitting the defining series of the Riemann zeta function at gives

where is the Hurwitz zeta function. The formula holds for by the series and extends by analytic continuation to . At the difference has a removable singularity. The multiple harmonic number extends to nested sums, and the hyperharmonic number uses repeated partial sums.

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

which relates to an indefinite version of a famous series for . also satisfies

where is the Riemann zeta function. This follows from the identity

where is the trigamma function since

For and odd , the generalized harmonic numbers have the explicit form

where is the polygamma function, is the gamma function, and is the Riemann zeta function.

The 2-index harmonic numbers satisfy the identity

(P. Simon, pers. comm., Aug. 30, 2004), where .

Sums of the generalized harmonic numbers include

for , where is a polylogarithm,

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

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

(M. Trott, pers. comm.).

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

where is real and

This gives the special results

for , , and , respectively, where is Catalan's constant.

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

the third-order harmonic number by

and the th-order harmonic number, for positive integer , by

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

plus the recurrence relation

for , where is the Roman symbol. For general and , this is equivalent to

and for , it simplifies to

For , the harmonic number can be written

where is the Roman factorial and 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 ." 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 and for Certain Values of and ." 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." [email protected] 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 ." 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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