Analytic Number Theory >> Content Detail



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This course is an introduction to analytic number theory, including the use of zeta functions and L-functions to prove distribution results concerning prime numbers (e.g., the prime number theorem in arithmetic progressions). There is also a bit of discussion of non-abelian equidistribution results, like the Chebotarev density theorem, and the Sato-Tate conjecture for elliptic curves (and recent progress on same by Clozel, Harris, Shepherd-Barron, Taylor). For a detailed description of the content and structure of the course, please see the first set of lecture notes. (PDF)


There is no required text for this class. Detailed lecture notes are provided in lieu of following a specific text.

Recommended Texts

Amazon logo Davenport, H., and H. L. Montgomery. Multiplicative Number Theory. New York, NY: Springer-Verlag, 2000. ISBN: 9780387950976.

Amazon logo Iwaniec, Henryk, and Emmanuel Kowalski. Analytic Number Theory. Providence, RI: American Mathematical Society, 2004. ISBN: 9780821836330.


Complex analysis (18.112), some background in number theory (at the level of 18.781). There are one or two points where a little algebraic number theory (as in 18.786) may be helpful, but it is in no way required. On the other hand, the 18.112 prerequisite is quite serious; if you do not formally meet it, you must let me know in writing what your equivalent preparation is. Also recommended: some abstract algebra (18.701 and 18.702), though this will probably only make a real difference at the end of the semester.


There will be roughly weekly assignments throughout the semester.

Collaboration policy

You may (and should) work together on problems, but you must write up solutions individually, and you should indicate on your homework who you were working with. In case of ambiguity, I reserve the right to ask you to defend your solutions individually.


There are no exams in this class.


100% homework. This is a graduate course, after all, albeit one which is probably suitable for a sufficiently prepared and motivated undergraduate.


1Introduction to the course; the Riemann zeta function, approach to the prime number theorem
2Proof of the prime number theorem
3Dirichlet series, arithmetic functions
4Dirichlet characters, Dirichlet L-seriesProblem set 1 due
5Nonvanishing of L-series on the line Re(s)=1
6Dirichlet and natural density, Fourier analysis; Dirichlet's theorem
7Prime number theorem in arithmetic progressions; functional equation for zetaProblem set 2 due
8Functional equation for zeta (cont.)
9Functional equations for Dirichlet L-functions
10Error bounds in the prime number theorem; the Riemann hypothesisProblem set 3 due

Zeroes of zeta in the critical strip; a zero-free region

12A zero-free region; von Mangoldt's formula
13von Mangoldt's formula (cont.)Problem set 4 due
14von Mangoldt's formula; error bounds in arithmetic progressions
15Error bounds in arithmetic progressions (cont.)
16Introduction to sieve methods: the sieve of Eratosthenes
17Guest lecture by Professor Ben GreenProblem set 5 due
18The sieve of Eratosthenes (cont.); Brun's combinatorial sieve
19Brun's combinatorial sieve (cont.)
20The Selberg sieveProblem set 6 due
21The Selberg sieve (cont.); applying the Selberg sieve
22Introduction to large sieve inequalities
23A multiplicative large sieve inequality; an application of the large sieveProblem set 7 due
24The Bombieri-Vinogradov theorem (statement)
25The Bombieri-Vinogradov theorem (proof)Problem set 8 due
26The Bombieri-Vinogradov theorem (proof, cont.)
27The Bombieri-Vinogradov theorem (proof, cont.); prime k-tuplesProblem set 9 due
28Short gaps between primes
29Short gaps between primes (cont.)
30Short gaps between primes (proofs)Problem set 10 due
31Short gaps between primes (proofs, cont.)
32Short gaps between primes (proofs, cont.)
33Artin L-functions and the Chebotarev density theoremProblem set 11 due
34Artin L-functions
35Equidistribution in compact groups
36Elliptic curves; the Sato-Tate distribution


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