Dirichlet Series

Manuel Eberl 🌐

October 12, 2017

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
These results are useful as a basis for developing more number-theoretic results, such as the Prime Number Theorem.
BSD License

Topics

Theories of Dirichlet_Series