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

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

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

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

Uses

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

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

Wherer ${\displaystyle \gamma }$ is the Euler–Mascheroni constant.

See also

• Divisor summatory function

References