Sequences of Prime Reciprocals. Preliminaries

In the article we formalize some properties needed to prove that sequences of prime reciprocals are divergent. The aim is to show that the series exhibits log-log growth. We introduce some auxiliary notions as harmonic numbers, telescoping series, and prove some standard properties of logarithms and exponents absent in the Mizar Mathematical Library. At the end we proceed with square-free and square-containing parts of a natural number and reciprocals of corresponding products.Institute of Informatics, University of Białystok, PolandGrzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pąk. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definition of integrability for partial functions from ℝ to ℝ and integrability for continuous functions. Formalized Mathematics, 9(2):281–284, 2001.Noboru Endou, Yasunari Shidama, and Masahiko Yamazaki. Integrability and the integral of partial functions from ℝ into ℝ. Formalized Mathematics, 14(4):207–212, 2006. doi:10.2478/v10037-006-0023-y.Leonhard Euler. Variae observationes circa series infinitas. Commentarii Academiae Scientiarum Petropolitanae, 9:160–188, 1737.Adam Grabowski and Christoph Schwarzweller. On duplication in mathematical repositories. In Serge Autexier, Jacques Calmet, David Delahaye, Patrick D. F. Ion, Laurence Rideau, Renaud Rioboo, and Alan P. Sexton, editors, Intelligent Computer Mathematics, 10th International Conference, AISC 2010, 17th Symposium, Calculemus 2010, and 9th International Conference, MKM 2010, Paris, France, July 5–10, 2010. Proceedings, volume 6167 of Lecture Notes in Computer Science, pages 300–314. Springer, 2010. doi:10.1007/978-3-642-14128-7_26.Adam Grabowski and Christoph Schwarzweller. Revisions as an essential tool to maintain mathematical repositories. In M. Kauers, M. Kerber, R. Miner, and W. Windsteiger, editors, Towards Mechanized Mathematical Assistants. Lecture Notes in Computer Science, volume 4573, pages 235–249. Springer: Berlin, Heidelberg, 2007.Artur Korniłowicz and Karol Pąk. Basel problem – preliminaries. Formalized Mathematics, 25(2):141–147, 2017. doi:10.1515/forma-2017-0013.Artur Korniłowicz and Piotr Rudnicki. Fundamental Theorem of Arithmetic. Formalized Mathematics, 12(2):179–186, 2004.Robert Milewski. Natural numbers. Formalized Mathematics, 7(1):19–22, 1998.Akira Nishino and Yasunari Shidama. The Maclaurin expansions. Formalized Mathematics, 13(3):421–425, 2005.Karol Pąk and Artur Korniłowicz. Basel problem. Formalized Mathematics, 25(2):149–155, 2017. doi:10.1515/forma-2017-0014.261697