Dirichlet Hyperbola Method

The Dirichlet hyperbola method is an ingenious technique for evaluating sums involving Dirichlet convolutions by exploiting the geometry of divisor pairs.

TheoremDirichlet Hyperbola Identity

Let $f$ and $g$ be arithmetic functions with summatory functions:

F(x) = \sum_{n \leq x} f(n), \quad G(x) = \sum_{n \leq x} g(n)

Then for $h = f * g$ :

\sum_{n \leq x} h(n) = \sum_{d \leq u} f(d) G(x/d) + \sum_{m \leq v} g(m) F(x/m) - F(u) G(v)

where $uv = x$ and typically we choose $u = v = \sqrt{x}$ .

ExampleDivisor Function Asymptotics

Taking $f = g = 1$ gives $h = \tau$ (the divisor function):

\sum_{n \leq x} \tau(n) = 2 \sum_{d \leq \sqrt{x}} \lfloor x/d \rfloor - \lfloor \sqrt{x} \rfloor^2

The main term comes from $\lfloor x/d \rfloor \approx x/d$ :

\sum_{n \leq x} \tau(n) \approx 2x \sum_{d \leq \sqrt{x}} \frac{1}{d} \approx 2x \log \sqrt{x} = x \log x

More precisely: $\sum_{n \leq x} \tau(n) = x \log x + (2\gamma - 1)x + O(\sqrt{x})$ .

ExamplePiltz Divisor Problem

For $\tau_k(n) = \sum_{d_1 \cdots d_k = n} 1$ (generalized divisor function), the hyperbola method extends to:

\sum_{n \leq x} \tau_k(n) = x P_{k-1}(\log x) + O(x^{1 - 1/k + \epsilon})

where $P_{k-1}$ is a polynomial of degree $k-1$ .

Remark

The geometric intuition: we count lattice points $(d,m)$ with $dm \leq x$ . The region below the hyperbola is divided into:

Points with $d \leq \sqrt{x}$ (vertical strips)
Points with $m \leq \sqrt{x}$ (horizontal strips)
Overlap at $d, m \leq \sqrt{x}$ (square), subtracted once

This "cut the hyperbola" approach appears throughout analytic number theory.

The hyperbola method transforms difficult summation problems into tractable pieces. Its power lies in reducing convolution sums to simpler summatory functions, often yielding both main terms and error bounds with geometric clarity.