Dirichlet hyperbola method

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

[math]\displaystyle{ \sum_{n \le x} f(n) }[/math]

where [math]\displaystyle{ f, g, h }[/math] are multiplicative functions with [math]\displaystyle{ f = g*h }[/math], where [math]\displaystyle{ * }[/math] is the Dirichlet convolution. It uses the fact that

[math]\displaystyle{ \sum_{n\leq x}f(n) = \sum_{n\leq x}\sum_{ab=n} g(a)h(b) = \sum_{a\leq\sqrt{x}}\sum_{b\leq\frac{x}{a}} g(a)h(b) + \sum_{b\leq\sqrt{x}}\sum_{a\leq\frac{x}{b}} g(a)h(b) - \sum_{a\leq\sqrt{x}}\sum_{b\leq\sqrt{x}} g(a)h(b). }[/math]

Uses

Let [math]\displaystyle{ \tau (n) }[/math] be the number-of-divisors function. Since [math]\displaystyle{ \tau = 1 * 1 }[/math], the Dirichlet hyperbola method gives us the result

[math]\displaystyle{ \sum_{n \le x} \tau (n) = x \log x + (2\gamma - 1)x + O(\sqrt x). }[/math]

Wherer [math]\displaystyle{ \gamma }[/math] is the Euler–Mascheroni constant.

See also

• Divisor summatory function