Description: Analytic number theory is a branch of number theory that uses techniques from analysis to solve problems about the integers. It is well known for its results on prime numbers (for example the celebrated Prime Number Theorem states that the number of prime numbers less than N is about N/logN) and additive number theory (the recently proved Goldbach’s weak conjecture states that every odd number greater than 7 can be expressed as the sum of three odd primes). Depending on specific interests, we can read any one of the following texts:

(1) “An Introduction to Analytic Number Theory” by Tom Apostol. A proof of Prime Number Theorem using analytic properties of the zeta function.

(2) “Additive Number Theory” by Melvyn Nathanson. This book highlights two important methods in additive number theory: sieve methods and the circle method. Both of these methods are proved to be very effective in obtaining results towards the twin prime conjecture and the Goldbach conjecture.

(3) Multiplicative Number Theory I. Classical Theory by Montgomery and Vaughan. A more advanced introduction to the Prime Number Theorem and primes in arithmetic progressions.

Term:

Spring

Prerequisites:

basic algebra(linear algebra, groups, rings, fields), basics of real and complex analysis

Year:

2015