# Nine Chapters of Analytic Number Theory in Isabelle/HOL (unpublished draft)

by Manuel Eberl

### Abstract:

In this paper, I present a formalisation of a large portion of Apostol's

Introduction to Analytic Number Theoryin Isabelle/HOL. Of the 14 chapters in the book, the content of 9 has been mostly formalised, while the content of 3 others was already mostly available in Isabelle before.The most interesting results that were formalised are:

- The Riemann and Hurwitz
ζfunctions and the DirichletLfunctions- Dirichlet's theorem on primes in arithmetic progressions
- An analytic proof of the Prime Number Theorem
- The asymptotics of arithmetical functions such as the prime
ωfunction, the divisor countσ, and Euler's totient function_{0}(n)φ(n)

### BibTeX:

```
@inproceedings{eberl19ant,
author="Eberl, Manuel",
title="Nine Chapters of Analytic Number Theory in {I}sabelle/{HOL}",
year="2019",
note="Draft available at \url{https://www21.in.tum.de/~eberlm/pdfs/ant.pdf}"
}
```