# 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, as y ↓ 0, the non-negative sequence an is such that there is an asymptotic equivalence

$\sum _{n=0}^{\infty }a_{n}e^{-ny}\sim {\frac {1}{y}}$ then there is also an asymptotic equivalence

$\sum _{k=0}^{n}a_{k}\sim n$ as n → ∞. 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 an ≥ 0 for all n, and as x ↑1 we have

$\sum _{n=0}^{\infty }a_{n}x^{n}\sim {\frac {1}{1-x}}.$

Then as n goes to ∞ we have

$\sum _{k=0}^{n}a_{k}\sim n.$

The theorem is sometimes quoted in equivalent forms, where instead of requiring an ≥ 0, we require an = O(1), or we require an ≥ −K for some constant K.:155 The theorem is sometimes quoted in another equivalent formulation (through the change of variable x = 1/ey ).:155 If, as y ↓ 0,

$\sum _{n=0}^{\infty }a_{n}e^{-ny}\sim {\frac {1}{y}}$

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,∞) → 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 ρ is a non-negative real number, then the following statements are equivalent

• $\omega (s)\sim Cs^{-\rho },\quad {\rm {{as\ }s\to 0}}$
• $F(t)\sim {\frac {C}{\Gamma (\rho +1)}}t^{\rho },\quad {\rm {{as\ }t\to \infty .}}$

Here Γ denotes the Gamma function. One obtains the theorem for series as a special case by taking ρ = 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 positive t. Let L be a function slowly varying at infinity and ρ a non-negative real number. Then the following statements are equivalent

• $\omega (s)\sim s^{-\rho }L(s^{-1}),\quad {\rm {{as\ }s\to 0}}$
• $F(t)\sim {\frac {1}{\Gamma (\rho +1)}}t^{\rho }L(t),\quad {\rm {{as\ }t\to \infty .}}$

## Karamata's proof

Karamata (1930) found a short proof of the theorem by considering the functions g such that

$\lim _{x\rightarrow 1}(1-x)\sum a_{n}x^{n}g(x^{n})=\int _{0}^{1}g(t)dt$

An easy calculation shows that all monomials g(x)=xk 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 an 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 Σanxng(xn) is a0+...+aN, 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 tends 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 an = O(1/n), and as x ↑ 1 we have

$\sum a_{n}x^{n}\to s,$

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 Λ 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