PlanetMath: convolution method

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convolution method

(Definition)

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

$\begin{displaymath}\begin{array}{ll} \displaystyle \sum_{n \le x} f(n) & \displa... ...= \sum_{d \le x} g(d) \sum_{m \le \frac{x}{d}} h(m) \end{array}\end{displaymath}$

This method for calculating $\displaystyle \sum_{n \le x} f(n)$ is advantageous when the sums in terms of and are easier to handle.

As an example, the sum $\displaystyle \sum_{n \le x} \mu^2(n)$ will be calculated using the convolution method.

Since $\mu^2=\mu^2*\varepsilon=\mu^2*(\mu*1)=(\mu^2*\mu)*1$ , the functions $\mu^2*\mu$ and can be used.

To use the convolution method, a nice way to calculate $(\mu^2*\mu)(n)$ needs to be found. Note that $\mu^2*\mu$ is multiplicative, so it only needs to be evaluated at prime powers.

Let $k \in \mathbb{N}$ . Then

$\displaystyle (\mu^2*\mu)(p^k) = \sum_{j=0}^k \mu^2(p^j) \mu(p^{k-j}) = \mu(p^k... ...}) = \begin{cases} 0 & \text{if } k \neq 2 \ -1 & \text{if } k=2. \end{cases}$

Since $\mu^2*\mu$ is multiplicative, then $(\mu^2*\mu)(n)= \begin{cases} 0 & \text{if } n \text{ is not a square} \ \mu(\sqrt{n}) & \text{if } n \text{ is a square.} \end{cases}$

The convolution method yields:

$\begin{displaymath}\begin{array}{ll} \displaystyle \sum_{n \le x} \mu^2(n) & \di... ... & \displaystyle = \frac{6}{\pi^2}x + O(\sqrt{x}) \end{array}\end{displaymath}$

"convolution method" is owned by Wkbj79.

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See Also: Dirichlet hyperbola method, Möbius function, $\displaystyle \sum_{n \le x} y^{\Omega(n)}=O\left( \frac{x(\log x)^{y-1}}{2-y} \right)$ for $1 \le y<2$ , $\displaystyle x\log^2x=O\left(\sum_{n \le x} 2^{\Omega(n)} \right)$

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Cross-references: prime, functions, sums, multiplicative functions
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This is version 10 of convolution method, born on 2006-06-10, modified 2007-05-30.
Object id is 7989, canonical name is ConvolutionMethod.
Accessed 2658 times total.

Classification:

AMS MSC:

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

Pending Errata and Addenda

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