# Hardy–Littlewood Tauberian theorem

In mathematical analysis, the Hardy–Littlewood Tauberian theorem is a Tauberian theorem relating the asymptotics of the partial sums of a series with the asymptotics of its Abel summation. In this form, the theorem asserts that if the sequence $a_{n}\geq 0$ is such that there is an asymptotic equivalence
    $\sum _{n=0}^{\infty }a_{n}e^{-ny}\sim {\frac {1}{y}}\ {\text{as}}\ y\downarrow 0$
then there is also an asymptotic equivalence
    $\sum _{k=0}^{n}a_{k}\sim n$
as $n\to \infty$. The integral formulation of the theorem relates in an analogous manner the asymptotics of the cumulative distribution function of a function with the asymptotics of its Laplace transform.
The theorem was proved in 1914 by G. H. Hardy and J. E. Littlewood.$^{:226}$ In 1930, Jovan Karamata gave a new and much simpler proof.$^{:226}$

## Statement of the theorem
### Series formulation

This formulation is from Titchmarsh.$^{:226}$ Suppose $a_{n}\geq 0$ for all $n\in \mathbb {N}$, and we have
    $\sum _{n=0}^{\infty }a_{n}x^{n}\sim {\frac {1}{1-x}}\ {\text{as}}\ x\uparrow 1.$
Then as $n\to \infty$ we have
    $\sum _{k=0}^{n}a_{k}\sim n.$
The theorem is sometimes quoted in equivalent forms, where instead of requiring $a_{n}\geq 0$, we require $a_{n}=O(1)$, or we require $a_{n}\geq -K$ for some constant $K$.$^{:155}$ The theorem is sometimes quoted in another equivalent formulation (through the change of variable $x=1/e^{y}$).$^{:155}$ If,
    $\sum _{n=0}^{\infty }a_{n}e^{-ny}\sim {\frac {1}{y}}\ {\text{as}}\ y\downarrow 0$
then
    $\sum _{k=0}^{n}a_{k}\sim n.$

### Integral formulation

The following more general formulation is from Feller.$^{:445}$ Consider a real-valued function $F:[0,\infty )\to \mathbb {R}$ of bounded variation. The Laplace–Stieltjes transform of $F$ is defined by the Stieltjes integral
    $\omega (s)=\int _{0}^{\infty }e^{-st}\,dF(t).$
The theorem relates the asymptotics of ω with those of $F$ in the following way. If $\rho$ is a non-negative real number, then the following statements are equivalent
* $\omega (s)\sim Cs^{-\rho },\quad {\rm {{as\ }s\to 0}}$
* $F(t)\sim {\frac {C}{\Gamma (\rho +1)}}t^{\rho },\ {\text{as}}\ t\to \infty .$
Here $\Gamma$ denotes the Gamma function. One obtains the theorem for series as a special case by taking $\rho =1$ and $F(t)$ to be a piecewise constant function with value $\textstyle {\sum _{k=0}^{n}a_{k}}$ between $t=n$ and $t=n+1$.
A slight improvement is possible. According to the definition of a slowly varying function, $L(x)$ is slow varying at infinity iff
    ${\frac {L(tx)}{L(x)}}\to 1,\quad x\to \infty$
for every $t>0$. Let $L$ be a function slowly varying at infinity and $\rho \geq 0$. Then the following statements are equivalent
* $\omega (s)\sim s^{-\rho }L(s^{-1}),\quad {\text{as}}\ s\to 0$
* $F(t)\sim {\frac {1}{\Gamma (\rho +1)}}t^{\rho }L(t),\quad {\text{as}}\ t\to \infty .$

## Karamata's proof

Karamata (1930) found a short proof of the theorem by considering the functions $g$ such that
    $\lim _{x\to 1}(1-x)\sum a_{n}x^{n}g(x^{n})=\int _{0}^{1}g(t)dt$
An easy calculation shows that all monomials $g(x)=x^{k}$ have this property, and therefore so do all polynomials $g$. This can be extended to a function $g$ with simple (step) discontinuities by approximating it by polynomials from above and below (using the Weierstrass approximation theorem and a little extra fudging) and using the fact that the coefficients $a_{n}$ are positive. In particular the function given by $g(t)=1/t$ if $1/e<t<1$ and $0$ otherwise has this property. But then for $x=e^{-1/N}$ the sum $\sum a_{n}x^{n}g(x^{n})$ is $a_{0}+\cdots +a_{N}$ and the integral of $g$ is $1$, from which the Hardy–Littlewood theorem follows immediately.

## Examples
### Non-positive coefficients

The theorem can fail without the condition that the coefficients are non-negative. For example, the function
    ${\frac {1}{(1+x)^{2}(1-x)}}=1-x+2x^{2}-2x^{3}+3x^{4}-3x^{5}+\cdots$
is asymptotic to $1/4(1-x)$ as $x\to 1$, but the partial sums of its coefficients are 1, 0, 2, 0, 3, 0, 4, ... and are not asymptotic to any linear function.

### Littlewood's extension of Tauber's theorem

In 1911 Littlewood proved an extension of Tauber's converse of Abel's theorem. Littlewood showed the following: If $a_{n}=O(1/n)$, and we have
    $\sum a_{n}x^{n}\to s\ {\text{as}}\ x\uparrow 1$
then
    $\sum a_{n}=s.$
This came historically before the Hardy–Littlewood Tauberian theorem, but can be proved as a simple application of it.$^{:233–235}$

### Prime number theorem

In 1915 Hardy and Littlewood developed a proof of the prime number theorem based on their Tauberian theorem; they proved
    $\sum _{n=2}^{\infty }\Lambda (n)e^{-ny}\sim {\frac {1}{y}},$
where $\Lambda$ is the von Mangoldt function, and then conclude
    $\sum _{n\leq x}\Lambda (n)\sim x,$
an equivalent form of the prime number theorem.$^{:34–35}$$^{:302–307}$ Littlewood developed a simpler proof, still based on this Tauberian theorem, in 1971.$^{:307–309}$