Dirichlet Series

# Dirichlet Series

 Definition: Let $(a_n)$ be a sequence (arithmetic function). The corresponding Dirichlet Series is the series is the function $\displaystyle{A(s) = \sum_{n=1}^{\infty} \frac{f(n)}{n^s}}$.

If $(a_n) = (a)$ is the constant sequence $a$ where $a \in \mathbb{C}$ then the corresponding Dirichlet series is:

(1)
\begin{align} \quad A(s) = \sum_{n=1}^{\infty} \frac{a}{n^s} \end{align}

Of course we can consider more complicated Dirichlet series. For example, consider the arithmetic function $\iota : N \to \mathbb{R}$ defined for all $n \in \mathbb{N}$ by $\iota (1) = 1$ and $\iota (n) = 0$ for all $n \geq 2$. Then the corresponding Dirichlet series is:

(2)
\begin{align} \quad B(s) = \sum_{n=1}^{\infty} \frac{\iota (n)}{n^s} = \frac{1}{1^s} + \frac{0}{2^s} + \frac{0}{3^s} + ... = 1 \end{align}
 Definition 1: Let $\displaystyle{A(s) = \sum_{m=1}^{\infty} \frac{a_m}{m^s}}$ and $\displaystyle{B(s) = \sum_{n=1}^{\infty} \frac{b_n}{n^s}}$ be two Dirichlet series. Then the Product of these two Dirichlet series is the Dirichlet series $\displaystyle{C(s) = \sum_{k=1}^{\infty} \frac{(a * b)(k)}{k^s}}$ where $a * b$ is the Dirichlet convolution of $a = (a_n)$ and $b = (b_n)$.

Recall that the Dirichlet convolution of $a = (a_n)$ and $b = (b_n)$ is $\displaystyle{(a * b)(k) = \sum_{n \mid k} a(n) b \left ( \frac{k}{n} \right )}$.