M391C: THE DISTRIBUTION OF PRIME NUMBERS
Instructor. Angel V. Kumchev; kumchev@math.utexas.edu.
Text. There will be no required text. I will provide lecture notes that draw heavily (but not excusively) on
The Distribution of Prime Numbers by M. N. Huxley (Oxford University Press, 1972) and Multiplicative
Number Theory, 3rd ed., by H. Davenport (Springer-Verlag, 2000). That being said, if you insist on having
a textbook or merely want to own a book on the subject, I strongly recommend Davenport’s book.
Prerequisites. Modest:
From Analysis: Working knowledge of real and complex analysis on the undergraduate level. Advanced
topics in complex function theory such as entire functions or the properties of the Γ-function will be reviewed
on the fly.
From Number Theory: Only the most basic arithmetic facts regarding divisibility and congruences will be
assumed. For example, the non-discretionary curriculum for M328K should more than suffice.
Description. This is an introductory course to Analytic Number Theory. It will acquaint the student with
some of the standard machinery in the field via examples arising from the investigation of the asymptotic
properties of prime numbers. Some of the main problems in the field that we will draw motivation from are:
• The Prime Number Theorem, which tells us how many primes p are there in the range 2 ≤ p ≤ x
when x is “large”.
• Dirichlet’s theorem on primes in arithmetic progressions, which tells us that any arithmetic progression a, a + q, a + 2q, . . . such that gcd(a, q) = 1 contains infinitely many primes.
• The centuries-old conjecture that for any two consecutive squares, there is a prime number that lies
between them.
These questions and their natural generalizations and reformulations will lead us to the discussion of topics such as: the Riemann zeta-function, Dirichlet characters and L-functions, the large sieve, approximate
functional equations for L-functions, and sieve identities.
Topics.
1. The Prime Number Theorem.
2. The Siegel–Walfisz theorem on primes in arithmetic progressions.
3. The large sieve.
4. Zero-density theorems.
5. Primes in short intervals.
6. The Bombieri–Vinogradov theorem.
7. Vinogradov’s three primes theorem.
8. Other topics as time permits.