Office: 456 Herman Brown Hall

Office Hours: 2--4 PM, T

Time: 1--2:15 PM, TR

Math 428/518

Instructor: Arindam Roy

Office: 456 Herman Brown Hall

Office Hours: 2--4 PM, T

Office: 456 Herman Brown Hall

Office Hours: 2--4 PM, T

Location: 423 Herman Brown Hall

Time: 1--2:15 PM, TR

Time: 1--2:15 PM, TR

**Course description::** In this course we deal with some problems and results in number theory that can be approached with tools of real and complex analysis. The most famous result of this type is the prime number theorem, which states that the number of less ot equal to $x$ is asymptotically equal to x/ log x. This is a result that the great number theorists of the 19th century and before believed to be true, but were unable to prove. A proof was finally given at the end of the 19th century by Hadamard and de la Vallée Poussin, using a new analytic approach, based on ideas of Bernhard Riemann.

A proof of the prime number theorem will form the core and highlight of this course. Other topics include the behavior of so-called arithmetic functions, and some results related to the Riemann zeta-function.

**Basic Plan:**
1) Arithmetic functions: theory of multiplicative and additive functions and averages and order of magnitude estimates.

2) Elementary theory of the distribution of primes.

3) Dirichlet series and Euler products.

4) Analytic methods for the distribution of primes. Theory of the Riemann Zeta function; connection between zeros of the zeta function and primes; analytic proof of the Prime Number Theorem.

5) Dirichlet's theorem on primes in arithmetic progressions.

6) As time permit a brief preview of some zeros distribution result of the Riemann zeta-function.

**Text:**
1) T. Apostol, "Introduction to Analytic Number Theory"

2) H. Davenport, "Multiplicative Number Theory"

3) H. L. Montgomery and R. C. Vaughan, "Multiplicative Number Theory I. Classical Theory"

4) E. C. Titchmarsh, "The Theory of the Riemann Zeta-Function"

**Grading:** The course grade will be based on few homework assignments and a presentation at the end of the semester.