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\title{Dirichlet Series}
\author{Manuel Eberl}
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\begin{abstract}
This entry is a formalisation of much of Chapters 2, 3, and 11 of Apostol's ``Introduction to Analytic Number Theory''~\cite{apostol}. This includes:
\begin{itemize} 
	\item Definitions and basic properties for several number-theoretic functions (Euler's $\varphi$, M\"{o}bius $\mu$, Liouville's $\lambda$, the divisor function $\sigma$, von Mangoldt's $\Lambda$)
\item Executable code for most of these functions, the most efficient implementations using the factoring algorithm by Thiemann\ \emph{et al.}
\item Dirichlet products and formal Dirichlet series
\item Analytic results connecting convergent formal Dirichlet series to complex functions
\item Euler product expansions
\item Asymptotic estimates of number-theoretic functions including the density of squarefree integers and the average number of divisors of a natural number
\end{itemize}
These results are useful as a basis for developing more number-theoretic results, such as the Prime Number Theorem.
\end{abstract}

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