Harmonic number

Harmonic number

The term harmonic number has multiple meanings. For other meanings, see harmonic number (disambiguation).

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

H_n= 1+frac{1}{2}+frac{1}{3}+cdots+frac{1}{n}
=sum_{k=1}^n frac{1}{k}.

This also equals n times the inverse of the harmonic mean of these natural numbers.

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 various expressions for various special functions.

Calculation

An integral representation is given by Euler:

H_n = int_0^1 frac{1 - x^n}{1 - x},dx.

This representation can be easily shown to satisfy the recurrence relation by the formula

int_0^1 x^n,dx = frac{1}{n + 1},

and then

x^{n} + frac{1 - x^n}{1 - x} = frac{1 - x^{n+1}}{1 - x}

inside the integral.

Hn grows about as fast as the natural logarithm of n. The reason is that the sum is approximated by the integral

int_1^n {1 over x}, dx

whose value is ln(n). More precisely, we have the limit:

lim_{n to infty} H_n - ln(n) = gamma

(where γ is the Euler-Mascheroni constant 0.5772156649dots), and the corresponding asymptotic expansion:

H_n = gamma + ln{n} + frac{1}{2}n^{-1} - frac{1}{12}n^{-2} + frac{1}{120}n^{-4} + mathcal{O}(n^{-6})

Special values for fractional arguments

There are the following special analytic values for fractional arguments between 0 and 1, given by the integral
H_alpha = int_0^1frac{1-x^alpha}{1-x},dx
More may be generated from the recurrence relation H_alpha = H_{alpha-1}+frac{1}{alpha}.

H_{1/2} = 2 -2ln{2},

H_{1/3} = 3-frac{pi}{2sqrt{3}} -frac{3}{2}ln{3}

H_{1/4} = 4-frac{pi}{2} - 3ln{2}

H_{1/6} = 6-frac{pi}{2}sqrt{3} -2ln{2} -frac{3}{2}ln(3)

H_{1/8} = 8-frac{pi}{2} - 4ln{2} - frac{1}{sqrt{2}} left{pi + ln(2 + sqrt{2}) - ln(2 - sqrt{2})right}

Generating functions

A generating function for the harmonic numbers is

sum_{n=1}^infty z^n H_n =
frac {-ln(1-z)}{1-z},

where ln(z) is the natural logarithm. An exponential generating function is

sum_{n=1}^infty frac {z^n}{n!} H_n =
-e^z sum_{k=1}^infty frac{1}{k} frac {(-z)^k}{k!} = e^z mbox {Ein}(z)

where mbox{Ein}(z) is the entire exponential integral. Note that

mbox {Ein}(z) = mbox{E}_1(z) + gamma + ln z =
Gamma (0,z) + gamma + ln z,

where Gamma (0,z) is the incomplete gamma function.

Applications

The harmonic numbers appear in several calculation formulas, such as the digamma function:

psi(n) = H_{n-1} - 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 γ, using the limit introduced in the previous section, although

gamma = lim_{n rightarrow infty}{left(H_n - lnleft(n+{1 over 2}right)right)}

converges more quickly.

In 2002 Jeffrey Lagarias proved that the Riemann hypothesis is equivalent to the statement that

sigma(n) le H_n + ln(H_n)e^{H_n},
is true for every integer n ≥ 1 with strict inequality if n > 1; here σ(n) denotes the sum of the divisors of n.

See also Watterson estimator, Tajima's D, coupon collector's problem.

Generalization

Generalized harmonic numbers

The generalized harmonic number of order n of m is given by

H_{n,m}=sum_{k=1}^n frac{1}{k^m}.

Note that the limit as n tends to infinity exists if m > 1.

Other notations occasionally used include

H_{n,m}= H_n^{(m)} = H_m(n).

The special case of m=1 is simply called a harmonic number and is frequently written without the superscript, as

H_n= sum_{k=1}^n frac{1}{k}.

In the limit of nrightarrow infty, the generalized harmonic number converges to the Riemann zeta function

lim_{nrightarrow infty} H_{n,m} = zeta(m).

The related sum sum_{k=1}^n k^m occurs in the study of Bernoulli numbers; the harmonic numbers also appear in the study of Stirling numbers.

A generating function for the generalized harmonic numbers is

sum_{n=1}^infty z^n H_{n,m} =
frac {mbox{Li}_m(z)}{1-z},

where mbox{Li}_m(z) is the polylogarithm, and |z|<1. The generating function given above for m=1 is a special case of this formula.

Generalization to the complex plane

Euler's integral formula for the harmonic numbers follows from the integral identity

int_a^1 frac {1-x^s}{1-x} dx =
- sum_{k=1}^infty frac {1}{k} {s choose k} (a-1)^k

which holds for general complex-valued s, for the suitably extended binomial coefficients. By choosing a=0, this formula gives both an integral and a series representation for a function that interpolates the harmonic numbers and extends a definition to the complex plane. This integral relation is easily derived by manipulating the Newton series

sum_{k=0}^infty {s choose k} (-x)^k = (1-x)^s,

which is just the Newton's generalized binomial theorem. The interpolating function is in fact just the digamma function:

psi(s+1)+gamma = int_0^1 frac {1-x^s}{1-x} dx

where psi(x) is the digamma, and gamma is the Euler-Mascheroni constant. The integration process may be repeated to obtain

H_{s,2}=-sum_{k=1}^infty frac {(-1)^k}{k} {s choose k} H_k.

References

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