## 11 Feb Some results named after Godfrey Harold Hardy

1. Hardy-Littlewood conjecture for twin primes

Let ${\pi_2(x)}$ denote the number of primes ${p\leq x}$ such that ${p+2}$ is also prime. Define the twin prime constant ${C_2}$ as

$\displaystyle C_2=\underset{p\geq 3}{\prod_{p\hspace{0.1cm} prime}}\Bigg(1-\frac{1}{(p-1)^2}\Bigg)\approx 0.66016181584686957392781211001455577...$then,

$\displaystyle \pi_2(x)\sim 2C_2\frac{x}{\ln^2(x)}$.

2. Hardy-Littlewood circle method

In mathematics, the Hardy-Littlewood circle method is a technique of analytic number theory. It is named after G. H. Hardy and J. E. Littlewood, who developed it in a series of papers on Waring’s problem. The initial idea is usually attributed to the work of Hardy with Srinivasa Ramanujan a few years earlier, in 1916 and 1917, on the asymptotics of the partition function.
Let us define ${F(z)}$ as,

$\displaystyle F(z)=\sum_{k=0}^{\infty}a_kz^k$and let the radius of convergence of ${F(z)}$ be larger than 1, then using Hardy-Littlewood circle method, we have,

$\displaystyle a_n=\frac{1}{2\pi i}\int_{C}\frac{F(z)}{z^{n+1}}dz$where ${C}$ is the unit circle oriented counter-clockwise.

3. 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 ${y\rightarrow 0}$, the non-negative sequence ${a_n}$ is such that there is an asymptotic equivalence

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

$\displaystyle \sum _{k=0}^{n}a_{k}\sim n$as ${n\rightarrow\infty.}$
The theorem was proved in 1914 by G. H. Hardy and J. E. Littlewood.

4. Hardy-Ramanujan asymptotic formula

In number theory, the partition function ${p(n)}$ represents the number of possible partitions of a non-negative integer ${n}$. For instance, ${p(4)=5}$ because the integer 4 has the five partitions 1 + 1 + 1 + 1, 1 + 1 + 2, 1 + 3, 2 + 2, and 4.
An asymptotic expression for ${p(n)}$ is given by,

$\displaystyle p(n)\sim \frac{1}{4n\sqrt{3}}exp\Bigg(\pi\sqrt{\frac{2n}{3}}\Bigg)$. This asymptotic formula was first obtained by G. H. Hardy and Ramanujan in 1918 and independently by J. V. Uspensky in 1920.

5. Hardy-Ramanujan theorem

In mathematics, the Hardy-Ramanujan theorem, proved by G. H. Hardy and Srinivasa Ramanujan (1917), states that the normal order of the number ${\omega(n)}$ of distinct prime factors of a number ${n}$ is ${\ln(\ln(n))}$. Roughly speaking, this means that most numbers have about this number of distinct prime factors.
It can also be written as, if ${\psi(x)}$ tends steadily to infinity with ${x}$, then

$\displaystyle \ln(\ln(x))-\psi(x)\sqrt{\ln(\ln(x))}<\omega(n)<\ln(\ln(x))+\psi(x)\sqrt{\ln(\ln(x))}$for almost all numbers ${n.