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The Dirichlet hyperbola method evaluates a summatory <Dirichlet convolution> by splitting the lattice points $ab\leq x$ at $a,b\leq\sqrt x$. For the <divisor function>,
$$
\sum_{n\leq x}\tau(n)
=2\sum_{a\leq\sqrt x}\left\lfloor\frac xa\right\rfloor
-\lfloor\sqrt x\rfloor^2.
$$

Dirichlet hyperbola method

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 150 / 1 / b / Solution

Dirichlet hyperbola method

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Dirichlet hyperbola method

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