Analytic methods in number theory: The Riemann Zeta function and the distribution of prime numbers

Analytic number theory is a branch of number theory, which examines arithmetic phenomena using the real and complex analysis. The most important in the topic is the analysis of prime numbers, the distribution of which contains profound regularities and irregularities, which have puzzled mathematicians over the centuries. The current paper gives a thorough and strict description of analytic techniques in number theory, along with the Riemann zeta function and its use in the distribution of prime numbers. We start with classical proofs of results on primes and arithmetic functions to derive the analytic theory of the zeta function, its analytic continuation, functional equation, and Euler product representation. We then look at the relationship between the zeta function with the prime distribution which eventually leads to the Prime Number Theorem and its generalizations. Another aspect of number theory that was covered in the paper is zero-free regions, explicit formulas, and the Riemann Hypothesis, and their far-reaching consequences. It is presented as a full self-contained exposition, which can be published in a peer-reviewed journal.