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. In 1930, Jovan Karamata gave a new and much simpler proof.

Series formulation
This formulation is from Titchmarsh. 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 $$K$$. The theorem is sometimes quoted in another equivalent formulation (through the change of variable $$x=1/e^y$$). 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. 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 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 $$t=n+1$$.
 * $$\omega(s)\sim C s^{-\rho},\quad\rm{as\ }s\to 0$$
 * $$F(t)\sim \frac{C}{\Gamma(\rho+1)}t^\rho, \ \text{as}\ t\to\infty.$$

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
found a short proof of the theorem by considering the functions $$g$$ such that
 * $$\lim_{x\to 1} (1-x)\sum a_nx^ng(x^n) = \int_0^1 g(t)dt$$

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.

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.

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
 * $$ \sum_{n \le x} \Lambda(n) \sim x,$$

an equivalent form of the prime number theorem. Littlewood developed a simpler proof, still based on this Tauberian theorem, in 1971.