**This is a development version of this entry. It might change over time and is not stable. Please refer to release versions for citations.**

### Abstract

This entry is a formalisation of much of Chapters 2, 3, and 11 of
Apostol's “Introduction to Analytic Number
Theory”. This includes:

- Definitions and basic properties for several number-theoretic functions (Euler's φ, Möbius μ, Liouville's λ, the divisor function σ, von Mangoldt's Λ)
- Executable code for most of these
functions, the most efficient implementations using the factoring
algorithm by Thiemann
*et al.* - Dirichlet products and formal Dirichlet series
- Analytic results connecting convergent formal Dirichlet series to complex functions
- Euler product expansions
- Asymptotic estimates of number-theoretic functions including the density of squarefree integers and the average number of divisors of a natural number