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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

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A Dirichlet series is a series of the form
$$
F(s)=\sum_{n=1}^{\infty}\frac{a_n}{n^s}.
$$
Multiplication of absolutely convergent Dirichlet series corresponds to <Dirichlet convolution> of their coefficients.

Dirichlet series

Let $\mathbf1(n)=1$. The basic <Von Mangoldt divisor identity> is
$$
\log n=\sum_{d\mid n}\Lambda(d),
$$
because if $n=\prod_pp^{v_p(n)}$, the right-hand side is $\sum_pv_p(n)\log p=\log n$. In terms of <Dirichlet convolution>, this says $\log=\mathbf1*\Lambda$. Since the <Möbius function> is the convolution inverse of $\mathbf1$, convolving with $\mu$ gives $\Lambda=\mu*\log$. Consequently
$$
\Lambda(n)=\sum_{d\mid n}\mu(d)\log\frac nd.
$$

Solved by gpt-5.6-sol high.

Solution

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

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The Dirichlet convolution of arithmetic functions $f$ and $g$ is
$$
(f*g)(n)=\sum_{d\mid n}f(d)g(n/d).
$$
It is associative and commutative, its identity is the function supported at $1$, and the <Möbius function> is the convolution inverse of the constant-one function.

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 Incoming links: Dirichlet convolution