Harmonic number

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The harmonic number with (red line) with its asymptotic limit (blue line) where is the Euler–Mascheroni constant.

In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers:

Starting from n = 1, the sequence of harmonic numbers begins:

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[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 regarding the long tail and the theory of network value.

The Bertrand-Chebyshev theorem implies that, except for the case n = 1, the harmonic numbers are never integers.[2]

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 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[edit]

There are several infinite summations involving harmonic numbers and powers of π:[3][better source needed]


An integral representation given by Euler[4] is

The equality above is straightforward by the simple algebraic identity

Using the substitution x = 1 − u, another expression for Hn is

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 is
where Bk 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. The exponential integral may also be expressed as
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. More precisely,

with some odd integers and .

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 defined harmonic primes to be the primes 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 .


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:
converges more quickly.

In 2002, Jeffrey Lagarias proved[11] 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 , and the corresponding eigenfunctions are given by the Legendre polynomials .[12]


Generalized harmonic numbers[edit]

The nth generalized harmonic number of order m is given by

(In some sources, this may also be denoted by or )

The special case m = 0 gives The special case m = 1 reduces to the usual harmonic number:

The limit of 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 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, ... (sequence A128670 in the OEIS)

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

where A is Apéry's constant ζ(3), and

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

  for example:

A generating function for the generalized harmonic numbers is

where is the polylogarithm, and |z| < 1. The generating function given above for m = 1 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:

where is the Riemann zeta function. The relevant recurrence relation is
Some special values are
where G is Catalan's constant,

In the special case that , we get

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

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 function, 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 ( is the Riemann zeta function).

Alternative, asymptotic formulation[edit]

When seeking to approximate Hx for a complex number x, it is effective to first compute Hm for some large integer m. Use that as an approximation for the value of Hm+x. 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 a fixed integer n, it is the case that

If n 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 n is replaced by an arbitrary complex number x,

Swapping the order of the two sides of this equation and then subtracting them from Hx gives

This infinite series converges for all complex numbers 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 this construction, the function that defines the harmonic number for complex values is the unique function that simultaneously satisfies (1) H0 = 0, (2) Hx = Hx−1 + 1/x for all complex numbers x except the non-positive integers, and (3) limm→+∞ (Hm+xHm) = 0 for all complex values x.

This last formula can be used to show 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 that

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

See also[edit]


  1. ^ a b John H., Conway; Richard K., Guy (1995). The book of numbers. Copernicus.
  2. ^ Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren (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 978-1-58488-347-0.
  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. 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 (4): 287–302. CiteSeerX doi:10.1080/10586458.1994.10504298.
  9. ^ Sanna, Carlo (2016). "On the p-adic valuation of harmonic numbers" (PDF). Journal of Number Theory. 166: 41–46. doi:10.1016/j.jnt.2016.02.020. hdl:2318/1622121.
  10. ^ Chen, Yong-Gao; Wu, Bing-Ling (2017). "On certain properties of harmonic numbers". Journal of Number Theory. 175: 66–86. doi:10.1016/j.jnt.2016.11.027.
  11. ^ Jeffrey Lagarias (2002). "An Elementary Problem Equivalent to the Riemann Hypothesis". Amer. Math. Monthly. 109 (6): 534–543. arXiv:math.NT/0008177. doi:10.2307/2695443. JSTOR 2695443.
  12. ^ E.O. Tuck (1964). "Some methods for flows past blunt slender bodies". J. Fluid Mech. 18 (4): 619–635. Bibcode:1964JFM....18..619T. doi:10.1017/S0022112064000453. S2CID 123120978.


External links[edit]

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