Math 205A [Fall 2018]
Analytic Number Theory

www.math.ucla.edu/~nandersen/205a

Nick Andersen
nandersen [at] math [dot] ucla [dot] edu
MS 5634

MWF 10:00 — 10:50 AM [MS 5118]

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Here are the Lecture Notes

By appointment; just send me an email.

This course is mainly focused on studying the Riemann zeta function and Dirichlet L-functions from the analytic perspective. Our main tools will be complex analysis and cleverness. Among other things, we will investigate the Euler product, analytic continuation, functional equation, and explicit formulas for these functions and use them to attempt to answer questions about prime numbers.

Complex Analysis: either Math 246A or an A grade in Math 132

There is no required textbook, but the following references might be useful:
Davenport, Multiplicative Number Theory
Apostol, Introduction to Analytic Number Theory
Hildebrand, ANT Course Notes

There will be three homework assignments, collected in lecture every third Friday beginning 19 October.

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Feel free to work in groups on the homework assignments, but each person should write up their solutions individually.

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Here are your assignments:

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[HW1]  [HW2]  [HW3]

We will not have a final exam. Instead, each person will present a 10 minute talk during Week 10 with a 3 page LaTeX'd writeup of their presentation. Students may work in pairs if desired; simply double the length of the presentation and writeup. Here is an incomplete list of possible topics to present (if a student or group would like to present on a topic which is not on this list, they may do so with the instructor's permission). Topics should be decided by Monday of Week 6 (5 Nov 2018).

Primes:

Erdös and Selberg's elementary proof of the prime number theorem	The Erdös-Selberg dispute by Goldfeld
Newman's short proof of the prime number theorem	Newman's Short Proof by Zagier
Vinogradov's 3 primes theorem	Wikipedia: Vinogradov's Theorem and The Ternary Goldbach Conjecture is True by Helfgott
Chebyshev's bias	Wikipedia: Chebyshev's bias and Chebyshev's Bias by Rubinstein and Sarnak
Littlewood's Theorem	Wikipedia: Skewes's Number
The Mertens Conjecture	Wikipedia: Mertens Conjecture

The Riemann ζ function:

The number of zeros of ζ(s) on the critical line	A Short Proof of Levinson's Theorem by Young
The Lindelöf hypothesis for ζ(s)	Wikipedia: The Lindelöf Hypothesis
The approximate functional equation for ζ(s)	Chapter 4 of The Theory of the Riemann Zeta Function by Titchmarsh
Zero density estimates	Zero Density Estimates blog post by Tao

Degree 2 L-functions:

Modular form L-functions for SL(2,Z)	Notes on Modular Forms and L-functions by Sutherland
Elliptic curve L-functions and the modularity theorem	Notes on the L-function of an Elliptic Curve by Shurman

Miscellaneous:

Dirichlet's divisor problem	Wikipedia: Dirichlet's Divisor Problem
Dirichlet's class number formula for quadratic fields	Wikipedia: Class Number Formula

Your final grade will be computed from your homework [70%] and your final presentation [30%].