## Math 775: Topics in Analytic Number Theory

**Fall 2012, Section 1 **

### Time:

MWF 1:00 p.m- 2:00 pm.
### Place:

3096 East Hall
### Instructor:

Jeffrey Lagarias, 3086 East Hall, 763-1186,
lagarias@umich.edu

### Office hours:

TBA, 3086 East Hall

(Or by appointment: call or email me)
**Course homepage:** http://www.math.lsa.umich.edu/~lagarias/
Public/html/m775wi12.html

**Text (required):** H. Davenport,

*Multiplicative Number Theory. Second edition.*
Revised by H. Montgomery.
Springer-Verlag: New York 1980.

**Text (optional) ** H. L. Montgomery and R. Vaughan,

* Multiplicative Number Theory I. Classical Theory *
Cambridge Univ. Press 2006

**Text (optional) ** G. Tenenbaum,

* Analytic and Probabilistic Number Theory *
Cambridge Univ. Press 1995

**Prerequisites:** The equivalent of Math 675 (number theory)
and Math 596 (complex variables);

**From departmental course description:**

This course will continue Math 675. Topics will include
some real variables topics and complex variables topics.

Topics on the
real analysis side may include the Selberg Sieve, the Large Sieve,
the Brun-Titchmarsh Theorem, the Bombieri-Vinogradov theorem.

It may include the Goldston-Pintz-Yildirim result on small gaps
between primes.
On the complex analysis side it should cover some of Tate's thesis,

the explicit formula of prime number theory, special values of
L-function. Other possible topics include estimates for

moments of the Riemann zeta function on the critical line and the
analogy with random matrix theory.

(These topics exceed a
semester's worth so only a subset will be covered.)
The textbooks consulted will remain the same as the first semester.

Some
additional topics may be taken from Dan Bump's book on
automorphic forms and representations.

### Schedule:

Here is a current

Syllabus
It is ** incomplete** and **tentative**,
and will be superseded by later versions as the course evolves.
**Homework Assignments:**

Homework 1-part 1

Homework 1-part 2

Homework 2-part 1

Homework 2-part 2

**Grading:** Discussed in class. Possible final project.