Harmonic number

In mathematics, the -th harmonic number is the sum of the reciprocals of the first natural numbers:
$H_n= 1+\frac+\frac+\cdots+\frac =\sum_^n \frac.$

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

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.

Bertrand's postulate implies that, except for the case , the harmonic numbers are never integers.

Identities involving harmonic numbers

By definition, the harmonic numbers satisfy the recurrence relation
$H_ = H_ + \frac.$

The harmonic numbers are connected to the Stirling numbers of the first kind by the relation
$H_n = \frac\left$right

The functions
$f_n(x)=\frac(\log x-H_n)$
satisfy the property
$f_n'(x)=f_(x).$
In particular
$f_1(x)=x(\log x-1)$
is an integral of the logarithmic function.

The harmonic numbers satisfy the series identities
$\sum_^n H_k = (n+1) H_ - n$
and
$\sum_^n H_k^2 = (n+1)H_^2 - (2 n +1) H_n + 2 n.$
These two results are closely analogous to the corresponding integral results
$\int_0^x \log y \ d y = x \log x - x$
and
$\int_0^x (\log y)^2\ d y = x (\log x)^2 - 2 x \log x + 2 x.$

Identities involving

There are several infinite summations involving harmonic numbers and powers of :
$\sum_^\infty \frac=\frac\pi ^2$
$\sum_^\infty \frac=\frac\pi^4$
$\sum_^\infty \frac=\frac\pi^4$
$\sum_^\infty \frac=\frac\pi^4$

Calculation

An integral representation given by Euler is
$H_n = \int_0^1 \frac\,dx.$

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

Using the substitution , another expression for is
\begin
H_n &= \int_0^1 \frac\,dx = \int_0^1\frac\,du \\ pt&= \int_0^1\left \sum_^n(-1)^k \binom nk u^\right,du
= -\sum_^n (-1)^k\binom nk \int_0^1u^\,du \\ pt&= -\sum_^n(-1)^k\frac\binom nk .
\end

The th harmonic number is about as large as the natural logarithm of . The reason is that the sum is approximated by the integral
$\int_1^n \frac\, dx,$
whose value is .

The values of the sequence decrease monotonically towards the limit
$\lim_ \left(H_n - \ln n\right) = \gamma,$
where is the Euler–Mascheroni constant. The corresponding asymptotic expansion is
$\begin H_n &\sim \ln+\gamma+\frac-\sum_^\infty \frac\\ &=\ln+\gamma+\frac-\frac+\frac-\cdots, \end$
where are the Bernoulli numbers.

Generating functions

A generating function for the harmonic numbers is
$\sum_^\infty z^n H_n = \frac ,$
where ln(''z'') is the natural logarithm. An exponential generating function is
$\sum_^\infty \frac H_n = -e^z \sum_^\infty \frac \frac = e^z \operatorname(z)$
where Ein(''z'') is the entire exponential integral. The exponential integral may also be expressed as
$\operatorname(z) = \mathrm_1(z) + \gamma + \ln z = \Gamma (0,z) + \gamma + \ln z$
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. Indeed, using 2-adic valuation, it is not difficult to prove that for $n \ge 2$ the numerator of $H_n$ is an odd number while the denominator of $H_n$ is an even number. More precisely,
$H_n=\frac\frac$
with some odd integers $a_n$ and $b_n$.

As a consequence of Wolstenholme's theorem, for any prime number $p \ge 5$ the numerator of $H_$is divisible by $p^2$. Furthermore, Eisenstein proved that for all odd prime number $p$ it holds
$H_ \equiv -2q_p(2) \pmod p$
where $q_p(2) = (2^ -1)/p$ is a Fermat quotient, with the consequence that $p$ divides the numerator of $H_$ if and only if $p$ is a Wieferich prime.

In 1991, Eswarathasan and Levine defined $J_p$ as the set of all positive integers $n$ such that the numerator of $H_n$ is divisible by a prime number $p.$ They proved that
$\\subseteq J_p$
for all prime numbers $p \ge 5,$ and they defined ''harmonic primes'' to be the primes $p$ such that $J_p$ has exactly 3 elements.

Eswarathasan and Levine also conjectured that $J_p$ is a finite set for all primes $p,$ and that there are infinitely many harmonic primes. Boyd verified that $J_p$ is finite for all prime numbers up to $p = 547$ 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 $1/e$. Sanna showed that $J_p$ has zero asymptotic density, while Bing-Ling Wu and Yong-Gao Chen proved that the number of elements of $J_p$ not exceeding $x$ is at most $3x^$, for all $x \geq 1$.

Applications

The harmonic numbers appear in several calculation formulas, such as the digamma function
$\psi(n) = H_ - \gamma.$
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:
$\gamma = \lim_,$
although
$\gamma = \lim_$
converges more quickly.

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

The eigenvalues of the nonlocal problem
$\lambda \varphi(x) = \int_^ \frac \, dy$
are given by $\lambda = 2H_n$, where by convention $H_0 = 0$, and the corresponding eigenfunctions are given by the Legendre polynomials $\varphi(x) = P_n(x)$.

Generalizations

Generalized harmonic numbers

The generalized harmonic number of order ''m'' of ''n'' is given by
$H_=\sum_^n \frac.$

Other notations occasionally used include
$H_= H_n^ = H_m(n).$

The special case of ''m'' = 0 gives $H_= n.$ The special case of ''m'' = 1 is simply called a harmonic number and is frequently written without the ''m'', as
$H_n= \sum_^n \frac.$

The limit as is finite if , with the generalized harmonic number bounded by and converging to the Riemann zeta function
$\lim_ H_ = \zeta(m).$

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 $\sum_^n k^m$ 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
$\int_0^a H_ \, dx = a \frac -H_$
and
$\int_0^a H_ \, dx = a A - \frac H_,$ where ''A'' is Apéry's constant ''ζ''(3),
and
$\sum_^n H_=(n+1)H_- H_ \text m \geq 0 .$

Every generalized harmonic number of order ''m'' can be written as a function of harmonic numbers of order $m-1$ using
$H_ = \sum_^ \frac + \frac$   for example: $H_ = \frac + \frac + \frac + \frac$

A generating function for the generalized harmonic numbers is
$\sum_^\infty z^n H_ = \frac ,$
where $\operatorname_m(z)$ 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 $p,q>0$ integer, and $m>1$ integer or not, we have from polygamma functions:
$H_=\zeta(m)-p^m\sum_^\infty \frac$
where $\zeta(m)$ is the Riemann zeta function. The relevant recurrence relation is
$H_=H_+\frac.$
Some special values are
$H_=16-8G-\tfrac\pi^2$ where ''G'' is Catalan's constant,
$H_=4-\tfrac$
$H_=8G+\tfrac-\tfrac\pi^2$
$H_=64-27\zeta(3)-\pi^3$
$H_=8-6\zeta(3)$
$H_=^3-27\zeta(3)+\pi^3$

In the special case that $p = 1$, we get
$H_=\zeta(m, 1) - \zeta(m, n+1),$
where $\zeta(m, n)$ 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 obtain
$H_=\frac\left(H_x+H_\right)+\ln 2,$
$H_=\frac\left(H_x+H_+H_\right)+\ln 3,$
or, more generally,
$H_=\frac\left(H_x+H_+H_+\cdots +H_ \right) + \ln n.$

For generalized harmonic numbers, we have
$H_=\frac\left(\zeta(2)+\frac\left(H_+H_\right)\right)$
$H_=\frac\left(6\zeta(2)+H_+H_+H_\right),$
where $\zeta(n)$ 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
$H_n^ = \frac1n.$
Then the nth hyperharmonic number of order ''r'' (''r>0'') is defined recursively as
$H_n^ = \sum_^n H_k^.$
In particular, $H_n^$ is the ordinary harmonic number $H_n$.

Harmonic numbers for real and complex values

The formulae given above,
$H_x = \int_0^1 \frac \, dt= -\sum_^\infty \frac$
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
$H_x = \psi(x+1)+\gamma,$
where is the digamma function, and is the Euler–Mascheroni constant. The integration process may be repeated to obtain
$H_=-\sum_^\infty \frac H_k.$

The Taylor series for the harmonic numbers is
$H_x=\sum_^\infty (-1)^\zeta (k)\;x^\quad\text , x, < 1$
which comes from the Taylor series for the digamma function ($\zeta$ is the Riemann zeta function).

Approximation using the Taylor series expansion

The harmonic number can be approximated using the first few terms of the Taylor series expansion:Claude Leibovici (https://math.stackexchange.com/users/82404/claude-leibovici), Harmonic series sum approximation, URL (version: 2018-11-11): https://math.stackexchange.com/q/2986766
$H_n = \gamma + \ln n + \frac + O\left(\frac\right) \simeq \gamma + \ln n + \frac$
where $\gamma = 0.57721...$ is the Euler–Mascheroni constant.

Alternative, asymptotic formulation

When seeking to approximate  for a complex number , it is effective to first compute  for some large integer . Use that to approximate a value for  and 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
\lim_ \left _ - H_m\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 .

\lim_ \left _ - H_m\right= 0,
Swapping the order of the two sides of this equation and then subtracting them from  gives

\beginH_x &= \lim_ \left _m - (H_-H_x)\right\\ pt&= \lim_ \left left(\sum_^m \frac\right) - \left(\sum_^m \frac\right) \right\\ pt&= \lim_ \sum_^m \left(\frac - \frac\right) = x \sum_^ \frac\, .
\end

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 .

Note that this last formula can be used to show that
$\int_0^1 H_x \, dx = \gamma,$
where  is the Euler–Mascheroni constant or, more generally, for every  we have:
$\int_0^nH_\,dx = n\gamma + \ln(n!) .$

Special values for fractional arguments

There are the following special analytic values for fractional arguments between 0 and 1, given by the integral
$H_\alpha = \int_0^1\frac\,dx\, .$

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

For example:
$H_ = 2 -2\ln$
$H_ = 3-\tfrac -\tfrac\ln$
$H_ = \tfrac(1-\ln)+\sqrt\tfrac$
$H_ = 4-\tfrac - 3\ln$
$H_ = \tfrac-3\ln+\tfrac$
$H_ = 6-\tfrac \sqrt -2\ln -\tfrac \ln$
$H_ = 8-\tfrac - 4\ln - \tfrac \left\$
$H_ = 12-3\left(\ln+\tfrac\right)-\pi\left(1+\tfrac\right)+2\sqrt\ln \left (\sqrt \right )$

For positive integers ''p'' and ''q'' with ''p'' < ''q'', we have:
$H_ = \frac +2\sum_^ \cos\left(\frac\right)\ln\left(\right)-\frac\cot\left(\frac\right)-\ln\left(2q\right)$

Relation to the Riemann zeta function

Some derivatives of fractional harmonic numbers are given by

\begin
\frac & = (-1)^n!\left zeta(n+1)-H_\right\\ pt\frac & = (-1)^(n+1)!\left zeta(n+2)-H_\right\\ pt\frac & = (-1)^\frac(n+2)!\left zeta(n+3)-H_\right
\end

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

\begin
H_x & = \sum_^\infty (-1)^x^n\zeta(n+1) \\ ptH_ & = \sum_^\infty (-1)^(n+1)x^n\zeta(n+2) \\ ptH_ & = \frac\sum_^\infty (-1)^(n+1)(n+2)x^n\zeta(n+3).
\end

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

\begin
H_ & = \frac\left(\zeta(2)-\frac\zeta(3)+\frac\zeta(4)-\frac \zeta(5) + \cdots\right) \\ ptH_ & = \frac\left(2\zeta(3)-\frac\zeta(4)+\frac\zeta(5)-\frac \zeta(6) + \cdots\right) \\ ptH_ & = \frac\left(2\cdot3\zeta(4)-\frac\zeta(5)+\frac\zeta(6)-\frac\zeta(7)+\cdots\right).
\end

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

Notes

References

*
*
* Ed Sandifer,
How Euler Did It — Estimating the Basel problem
'' (2003)
*
*
*

External links

*

{{DEFAULTSORT:Harmonic Number
Number theory