Prof: Andrew Granville
Bureau: 6153 André Aisenstadt,
Tel: 343-6583; Courriel:
andrew@dms.umontreal.ca
Book: Harold Davenport's "Multiplicative number theory", 3rd edn, Springer.
Classtimes: Monday 13h30-15h00;
Friday 11h-12h30, in Andre-Aisenstadt 4186.
Class dates: From Jan 8 to Apr 13, 2007
En Francais,
et page web complet
Course notes (regularly updated)
(Please be aware that these are "first drafts", and not
100% reliable. If in doubt see Davenport's book)
1. Infinitely many primes; non-analytic methods
2. Infinitely many primes; analytic methods
3. Infinitely many primes; complex analysis
4. Binary quadratic forms
7. Primer on analysis
8. Riemann's plan for proving the prime number theorem
9. The fundamental properties of the Riemann zeta function
10. The explicit formula and the proof of the prime number theorem
11. The prime number theorem for arithmetic progressions
13. The large sieve
16. Long gaps between primes
18. Short gaps between primes
References
Compulsory homework:
1.1d, 1.2a, 1.3d, 1.4e, 1.7a.
2.1b, 2.2a,b,c, 2.6a. 2.7c,d, 2.8a,b, 2.8b,d,
2.9a,b,c,d,e, 2.10a,b,c,d,e, 2.11a,b,c,d, 2.13a.
3.1a, 3.2a,f,g, 3.3a,b, 3.5a,b, 3,6b.
4.1a,b,c,f,g, 4.2a, 4.3a,b, 4.4b,d, 4.5a, 4.6c,d, 4.7a.
7.1a, 7.6a,b, 7.9a,b,c.
9.1a,b, 9.3a, 9.5a, 9.8a,b,c,d.
11.1a, 11.4a,b, 11.5a, 11.8a,b,
Papers to read for background
Analytic Number Theory, and lectures (in French) at the ENS Paris
It is easy to
determine whether a given integer is prime
Prime number races
Prime number patterns
This will be a first course in analytic number theory,
in which we will study many aspects of the distribution of primes,
including a proof of the prime number theorem, the Bombieri-Vinogradov
theorem, and a proof of the recent celebrated result of Goldston,
Pintz and Yildirim on short gaps between primes.
This course will be most suitable for students who have had
a first course in number theory, and perhaps a course in complex analysis.