📂Number Theory

Generalized Dirichlet Product Representation for partial Sums of Arithmetic Functions

Lemma ¹

Let’s define $h = f \ast g$ for the arithmetic function $f,g,h$ as follows: $$ F (x) := \sum_{n \le x} f(x) \\ G (x) := \sum_{n \le x} g(x) \\ H (x) := \sum_{n \le x} h(x) $$ Then, $$ H = f \circ G = g \circ F $$ where the operation $\circ$ refers to the generalized convolution. In other words, the following is true: $$ H(x) = \sum_{n \le x} f(n) G \left( {{ x } \over { n }} \right) = \sum_{n \le x} g(n) F \left( {{ x } \over { n }} \right) $$

Proof

$$ U(x) := \begin{cases} 0 &, 0 < x < 1 \\ 1 &, 1 \le x\end{cases} $$ If we define a function $U : \mathbb{R}^{+} \to \mathbb{C}$ such that it is in $x \in (0,1)$ to $U(x) = 0$ as follows, $$ F = f \circ U \\ G = g \circ U $$

Properties of Generalized Convolution: If $\alpha$ and $\beta$ are arithmetic functions and $F , G : \mathbb{R}^{+} \to \mathbb{C}$ is a function where the function value is $0$ in $x \in (0,1)$, then $$ \alpha \circ \left( \beta \circ F \right) = \left( \alpha \ast\ \beta \right) \circ F $$

According to the properties of generalized convolution, $$ f \circ G = f \circ \left( g \circ U \right) = \left( f \ast\ g \right) \circ U = H \\ g \circ F = g \circ \left( f \circ U \right) = \left( g \ast\ f \right) \circ U = H $$

■