# Dirichlet hyperbola method

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][2]

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

## References

1. ^ "Dirichlet hyperbola method". planetmath.org. Retrieved 2018-06-12.
2. ^ Tenenbaum, Gérald (2015-07-16). Introduction to Analytic and Probabilistic Number Theory. American Mathematical Soc. p. 44. ISBN 9780821898543.
Dirichlet hyperbola method

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