# Nine Chapters of Analytic Number Theory in Isabelle/HOL

by Manuel Eberl

In: *Proceedings of the 10th International Conference on Interactive Theorem Proving* (2019)

### DOI:

10.4230/LIPIcs.ITP.2019.16### 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)

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### BibTeX:

```
@inproceedings{eberl19ant,
author="Eberl, Manuel",
title="Nine Chapters of Analytic Number Theory in {I}sabelle/{HOL}",
year="2019",
publisher="Leibniz International Proceedings in Informatics",
booktitle = "10th International Conference on Interactive Theorem Proving (ITP 2019)",
pages = "16:1--16:19",
series = "Leibniz International Proceedings in Informatics (LIPIcs)",
ISBN = "978-3-95977-122-1",
ISSN = "1868-8969",
volume = "141",
editor = "John Harrison and John O'Leary and Andrew Tolmach",
publisher = "Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik",
address = {"Dagstuhl, Germany"
}
```