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Dirichlet hyperbola method (Definition)

Let $ f$, $ g$, and $ h$ be multiplicative functions such that $ f=g*h$, where $ *$ denotes the convolution of $ g$ and $ h$. The Dirichlet hyperbola method (typically shortened to hyperbola method) is a way to calculate $ \displaystyle \sum_{n \le x} f(n)$ by using the fact that $ f=g*h$:

$\displaystyle \sum_{n \le x} f(n) = \sum_{n \le x} \sum_{ab=n} g(a)h(b) = \sum_... ...le \frac{x}{b}} g(a)h(b) - \sum_{a \le \sqrt{x}} \sum_{b \le \sqrt{x}} g(a)h(b)$

Note that, since $ ab=n \le x$, not both of $ a$ and $ b$ can be larger than $ \sqrt{x}$. The Dirichlet hyperbola method follows from this fact as well as the inclusion-exclusion principle.

This method for calculating $ \displaystyle \sum_{n \le x} f(n)$ is advantageous when the sums in terms of $ g$ and $ h$ are easier to handle and when $ \vert g(n)-h(n)\vert$ is relatively small for most $ n \in \mathbb{N}$.

As an example, the sum $ \displaystyle \sum_{n \le x} \tau(n)$ will be calculated using the Dirichlet hyperbola method.

Note that $ \tau=1*1$. Thus:

\begin{displaymath}\begin{array}{ll} \displaystyle \sum_{n \le x} \tau(n) & \dis... ...laystyle = x \log x + (2 \gamma - 1)x + O(\sqrt{x}) \end{array}\end{displaymath}



"Dirichlet hyperbola method" is owned by Wkbj79.
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See Also: convolution method, $\tau$ function, Euler's constant

Other names:  hyperbola method
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Cross-references: sums, inclusion-exclusion principle, multiplicative functions
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This is version 11 of Dirichlet hyperbola method, born on 2006-06-10, modified 2007-06-28.
Object id is 7990, canonical name is DirichletHyperbolaMethod.
Accessed 1830 times total.

Classification:
AMS MSC: 11N37 (Number theory :: Multiplicative number theory :: Asymptotic results on arithmetic functions)
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