Dirichlet hyperbola method

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

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

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

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

Uses

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

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

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

See also

• Divisor summatory function