Dirichlet hyperbola method

In number theory, the Dirichlet hyperbola method is a technique to evaluate the sum

$\sum _{n\le x}f(n)$

where $f,g,h$ are multiplicative functions with $f=g*h$, where $*$ is the Dirichlet convolution. It uses the fact that

$\sum _{n\le x}f(n)=\sum _{n\le x}\sum _{ab=n}g(a)h(b)=\sum _{a\le \sqrt{x}}\sum _{b\le \frac{x}{a}}g(a)h(b)+\sum _{b\le \sqrt{x}}\sum _{a\le \frac{x}{b}}g(a)h(b)-\sum _{a\le \sqrt{x}}\sum _{b\le \sqrt{x}}g(a)h(b).$

Uses

Let $\tau (n)$ be the number-of-divisors function. Since $\tau =1*1$, the Dirichlet hyperbola method gives us the result

$\sum _{n\le x}\tau (n)=x\log x+(2\gamma -1)x+O(\sqrt{x}).$

Wherer $\gamma$ is the Euler–Mascheroni constant.

See also

• Divisor summatory function