The Fundamental Theorem Of Algebra And The Divergence Of Reciprocals Of Primes Looked At Through Bergman Spaces
Published 2013 · Mathematics
In this note, we present proofs for two well known facts in elementary number theory through Bergman spaces terminology. Both of these facts have complex analytic proofs (in addition to proofs by other techniques); therefore, it is not surprising to be able to write proofs by using Bergman spaces. However, we present these proofs here with hope for more connections between Bergman spaces and number theory. In the second section, before we talk about the proofs, we start with a brief introduction to Bergman spaces. In the third section, we present a proof of The Fundamental Theorem of Algebra (FTA). We refer to  for the history and six different proofs (including a complex analysis proof) of the FTA. There are more than 20 articles in MAA’s Monthly that present other proofs. The most recent one, , claims that at least 80 proofs exist in the literature and the note, , can be seen for comments on some of the proofs. In the fourth section, we look at the harmonic series of prime numbers. It is a well-known fact that the series of reciprocals of prime numbers diverges. This was first noted by Euler in 18th century. Section 33 of  and the references therein can be consulted to see how Euler did it. Two other rigorous proofs can be additionally found in the first section of  and on page 297 of . The one in  is due to Erdös. In the fourth section, we present another proof of this fact.