Harmonic number

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The harmonic number

{displaystyle H_{n,1}}

with

{displaystyle n=lfloor {x}rfloor }

(red label) with its asymptotic limit

{displaystyle gamma +log x}

(blue label).

In mathematics, the nth harmonic number is defined as the sum of the reciprocals of the first n natural numbers:

{displaystyle H_{n}=sum _{k=1}^{n}{frac {1}{k}}=1+{frac {1}{2}}+{frac {1}{3}}+cdots +{frac {1}{n}}}

This is also equal to n times the inverse of the harmonic mean.

Harmonic numbers have been studied since ancient times and are important in many branches of number theory. Sometimes loosely referred to as the harmonic series. They are closely related to the Riemann zeta function, and appear in various expressions for special functions.

Representation

The first representation, in integral form, was given by Leonhard Euler:

{displaystyle H_{n}=int _{0}^{1}{frac {1-x^{n}}{1-x}},dx.}

In this representation it is easy to show that a recursive relation is satisfied by the formula

{displaystyle int _{0}^{1}x^{n},dx={frac {1}{n+1}},}

and then

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

within the integral.

For natural numbers, H_n can also be represented as:

{displaystyle H_{n}=sum _{k=0}^{n-1}int _{0}^{1}x^{k},dx}

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

{displaystyle int _{1}^{n}{1 over x},dx}

whose value is log(n). Specifically, we have the following limit:

{displaystyle lim _{nto infty }H_{n}-log(n)=gamma }

(where γ is the Euler-Mascheroni constant 0.5772156649...).

And also, as the corresponding asymptotic expansion:

{displaystyle H_{n}=gamma +log {n}+{frac {1}{2}}n^{-1}-{frac {1}{12}}n^{-2}+{frac {1}{120}}n^{-4}+{mathcal {O}}(n^{-6})}

Generating functions

A generating function that indexes the harmonic numbers is

{displaystyle sum _{n=1}^{infty }z^{n}H_{n}={frac {-log(1-z)}{1-z}},}

where ${displaystyle log(z)}$ It's the natural logarithm. Another exponential generating function that indexes to harmonic numbers is:

{displaystyle 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 ${displaystyle {mbox{Ein}}(z)}$ is the whole exponential integral. Note that

{displaystyle {mbox{Ein}}(z)={mbox{E}}_{1}(z)+gamma +log z=Gamma (0,z)+gamma +log z,}

where ${displaystyle Gamma (0,z)}$ is the incomplete gamma function.

Applications

Harmonic numbers appear in various formulas for calculus expressions, such as this expression for the digamma function:

{displaystyle psi (n)=H_{n-1}-gamma.,}

This relation is also often used to define the extension of harmonic numbers to non-integer numbers n. Harmonic numbers are also used frequently to define γ, using the limit previously defined in the previous section, although

{displaystyle gamma =lim _{nrightarrow infty }{left(H_{n}-log left(n+{1 over 2}right)right)}}

it converges faster.

In 2001 Jeffrey Lagarias proved that the Riemann hypothesis is equivalent to saying that:

{displaystyle sigma (n)leq H_{n}+e^{H_{n}}log(H_{n}),}

is true for any integer n ≥ 1 with the strict inequality if n > 1; Here σ(n) denotes the sum of the divisors of n.

Generalizations

Generalized Harmonic Numbers

The generalized harmonic numbers of order n of m are given by the expression:

{displaystyle 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 are:

{displaystyle H_{n,m}=H_{n}^{(m)}=H_{m}(n)}

The special case of m = 1 is simply the nth harmonic number and is usually written without the upper index.

{displaystyle H_{n}=sum _{k=1}^{n}{frac {1}{k}}}

At the limit, when ${displaystyle nrightarrow infty }$ , the generalized harmonic numbers converge to the Riemann zeta function.

{displaystyle lim _{nrightarrow infty }H_{n,m}=zeta (m)}

Like in the sum ${displaystyle sum _{k=1}^{n}k^{m}}$ Bernoulli numbers appear, in the generalized harmonic numbers appear the Stirling numbers.

A generating function for generalized harmonic numbers is:

{displaystyle sum _{n=1}^{infty }z^{n}H_{n,m}={frac {{mbox{Li}}_{m}(z)}{1-z}}}

where ${displaystyle {mbox{Li}}_{m}(z)}$ It's the polylogaritmo, and $<math alttext="{displaystyle |z|日本語z日本語.1{displaystyle UDIz entails 1} <img alt="{displaystyle |z|$ . The generated function given above is a special case of this formula when m = 1.

Generalization to the complex plane

From Euler's integral formula for the harmonic numbers, the following identity is obtained:

{displaystyle 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 a general complex number s, using a suitable extension of the binomial coefficients. Choosing a = 0, this formula gives both representations (integral and series form) for a function that generates the harmonic numbers and extends the definition to the complex plane. This integral relation is easily obtained by manipulation of the Newton binomial:

{displaystyle sum _{k=0}^{infty }{s choose k}(-x)^{k}=(1-x)^{s}}

specifically, Newton's generalized binomial. The interpolated function is precisely the digamma function, like this:

{displaystyle psi (s+1)+gamma =int _{0}^{1}{frac {1-x^{s}}{1-x}}dx}

where ψ(x) is the digamma function, and γ is the Euler-Mascheroni constant. The integration process can be repeated to obtain

{displaystyle H_{s,2}=-sum _{k=1}^{infty }{frac {(-1)^{k}}{k}}{s choose k}H_{k}}

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