25 research outputs found
Computing the Mertens and Meissel-Mertens constants for sums over arithmetic progressions
We give explicit numerical values with 100 decimal digits for the Mertens
constant involved in the asymptotic formula for and, as a by-product, for the Meissel-Mertens constant
defined as , for , ...,
and .Comment: 12 pages, 6 table
On the constant in the Mertens product for arithmetic progressions. I. Identities
The aim of the paper is the proof of new identities for the constant in the
Mertens product for arithmetic progressions. We deal with the problem of the
numerical computation of these constants in another paper.Comment: References added, misprints corrected. 9 page
Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 B.C.--2017) and another new proof
In this article, we provide a comprehensive historical survey of 183
different proofs of famous Euclid's theorem on the infinitude of prime numbers.
The author is trying to collect almost all the known proofs on infinitude of
primes, including some proofs that can be easily obtained as consequences of
some known problems or divisibility properties. Furthermore, here are listed
numerous elementary proofs of the infinitude of primes in different arithmetic
progressions.
All the references concerning the proofs of Euclid's theorem that use similar
methods and ideas are exposed subsequently. Namely, presented proofs are
divided into 8 subsections of Section 2 in dependence of the methods that are
used in them. {\bf Related new 14 proofs (2012-2017) are given in the last
subsection of Section 2.} In the next section, we survey mainly elementary
proofs of the infinitude of primes in different arithmetic progressions.
Presented proofs are special cases of Dirichlet's theorem. In Section 4, we
give a new simple "Euclidean's proof" of the infinitude of primes.Comment: 70 pages. In this extended third version of the article, 14 new
proofs of the infnitude of primes are added (2012-2017
An annotated bibliography for comparative prime number theory
The goal of this annotated bibliography is to record every publication on the
topic of comparative prime number theory together with a summary of its
results. We use a unified system of notation for the quantities being studied
and for the hypotheses under which results are obtained. We encourage feedback
on this manuscript (see the end of Section~1 for details).Comment: 98 pages; supersedes "Comparative prime number theory: A survey"
(arXiv:1202.3408