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[1]

\sum _{n\leq 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

References

Tenenbaum, Gérald (2015-07-16). Introduction to Analytic and Probabilistic Number Theory. American Mathematical Soc. p. 44. ISBN 9780821898543.

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