**Text (required):** H. Davenport,
*Multiplicative Number Theory*, Second edition,
Revised by H. L. Montgomery,
Springer-Verlag: New York 1980

**Text (optional):** H. L. Montgomery and R. C. Vaughn,
* Multiplicative Number Theory: I. Classical Theory,*
Cambridge University Press, Cambridge 2007.

**Text (optional):** G. Tenenbaum,
* Introduction to Analytic and Probabilistic Number Theory,*
Cambridge University Press, Cambridge 1995.

** Text (optional):** H. Iwaniec and E. Kowalski,
* Analytic Number Theory,*
Colloquium Publications Vol. 53,
American Mathematical Society: Providence, RI 2004.

This syllabus is **incomplete** and **tentative**,
and will be superseded by later versions as the course evolves.
(Version March 14)
3[D, Chap. 21]

# | date | Refs: Sections in: Davenport[D]; Montgomery-Vaughan [MV],Tenenbaum [T] |

1 | 1/4 | Tate's Thesis. |

2 | 1/6 | Tate's Thesis (cont'd) |

3 | 1/9 | C.0 Tate's thesis: prehistory,
C.1 Hecke zeta function for ideal classes, C.2 Hecke L-functions, ray class characters C.3 Hecke L-function-grossencharacters |

4 | 1/11 | C.3 Real quadratic
grossencharacter, 1.0 Review Zeta fn-PNT, 12.4 Hadamard product for Dirichlet L-function, B(\chi). [D, chap. 12] |

5 | 1/13 | 14.1 Zero free region for L(s, \chi), complex characters [D. Chap. 14] |

| 1/16 | [Martin Luther King Day] |

6 | 1/18 | 14.2 Zero free region for
L(s, \chi), real characters, exceptional zero 14.3 Landau's theorem, 14.4 Unconditional upper bound for exceptional zero [D. Chap. 14] |

7 | 1/20 | 14.3 Landau exceptional zero bound, 14.4 Gap Principle, Page's theorem, 14.5 Unconditional Real zero bound |

8 | 1/23 | 16.1 Zeros of L-functions, N(T, \chi), 16.2 Low-lying zeros, N(T, \chi) for imprimitive characters [D. Chap. 16] |

9 | 1/25 | 16.3 Argument Principle Revisited, 19.1 Explicit Formula for L(s, \chi), 19.2 Proof outline [D. Chap. 19] |

10 | 1/27 | 19.3. Explicit Formula proof; 20.1 Prime Number Theorem in Arith. Prog. [D, Chap. 20] |

11 | 1/30 | 20.2 Refined Exceptional Zero Bound, 21.1 Siegel's Theorem Statment |

12 | 2/1 | 21.2 Siegel's Theorem- Key Lemma |

13 | 2/3 | 21.3 Siegel's Theorem Proof, 21.4 Discussion of Proof [D., Chap. 21] |

14 | 2/6 | 22.1 PNT in Arith. Prog. II,
22.2 Averages of Primes in AP's:
Bombieri-Vinogradov Thm statement; 23.1 Polya-Vinogradov inequality, prim chars., 23.2 Convex functions [D. Chap. 22, 23] |

15 | 2/8 | 23.3 Polya-Vinogradov inequality, general case, 27.1 Large Sieve, 27.2 Duality Principle [D, Chap. 23, 27] |

16 | 2/10 | 27.3 Approximate Identity Frame Constant, 27.4 Additive version of Large sieve [D, Chap. 27] |

17 | 2/13 | 27.4 Additive version of
Large sieve (cont'd) [D. Chap. 27], 27.5 Large sieve bound: Farey
Fractions, 27.6 Renyi's large sieve [D., Chap. 27] |

18 | 2/15 | 27.6 Renyi large sieve (cont'd),
27.7 Montgomery's refinement, 27.8 Linnik least quadratic nonresidue exceptional set [D., Chap. 27] |

19 | 2/17 | 27.9 Large Sieve: Multiplicative Form, 27.10 Large Sieve: Full Analytic Form, 27.11 Gallagher Lemma |

20 | 2/20 | 27.12 Proof of Analytic Large Sieve,
28.0 Bombieri-Vinogradov theorem outline, 28.1 Smoothed Bombieri-Vinogradov theorem: Preliminaries [parallel to D. Chap. 28] |

21 | 2/22 | Smoothed Bombieri-Vinogradov Theorem: 28.2 Basic Reductions, 28.3 The Mollifier, 28.4 Application of Large Sieve Estimates |

22 | 2/24 | Smoothed Bombieri-Vinogdradov:
28.4 Large Sieve estimates (cont) , 28.5 Mean square bounds for L-fns
(skipped); 28.6 Completion of smoothed B.-V. proof assuming 28.5 estimates. |

Feb. 25- Mar. 4 | [Winter Vacation] | |

23 | 3/5 | Smoothed Bombieri-Vinogradov:
Summary;
28.5 Mean-square bounds for L(s, \chi), L'(s, \chi) on critical line, 28.6 Completion of smoothed B.-V. proof (review) |

24 | 3/7 | 28.8 Application: Prime p with a prime divisor q of p-1 exceeding p^{0.6} (Goldfeld) |

25 | 3/9 | 28.7 Completion of Bombieri-Vinogradov theorem: unsmoothing |

26 | 3/12 | 29.1 Gallagher's mean square error for primes in arithmetic progressions [D., Chap. 29] |

27 | 3/14 | D.1 Lindelof's theorem, D.2 Vertical Growth of zeta function, D.3 Convexity bounds |

28 | 3/16 | E.1 Exponential sums, E.2 Weyl method: Linear and Quadratic Polynomials |

29 | 3/19 | E.2 Weyl method (continued). Presentation of Student Project Topics |

30 | 3/21 | E.3 Weyl method: higher degree case |

31 | 3/23 | E.4A Moments and size of zeta in critical strip, Conjectures, Random matrix view |

32 | 3/26 | E.4B. Moments conjecture, E.5 Subconvexity bound, finite sum part |

33 | 3/28 | E.5 Subconvexity bound, Finite sum part, E.6 Approximate functional equation |

34 | 3/30 | [No class] |

35 | 4/2 | E.6 Approximate function eqn, subconvexity, 26.0 Vinogradov sum of three primes outline |

36 | 4/4 | 26.0 Vinogradov sum of three primes outline (cont'd), 26.1 Major arcs estimate: Singular series |

37 | 4/6 | 26.1 Major arcs, singular series (cont.), 26.2 Minor arcs estimate |

38 | 4/9 | F.1 Hardy-Littlewood prime k-tuples, singular series, F.2 Uniform H-L conjecture, Gallagher's theorem, |

39 | 4/11 | F.3 Poisson and Exponential Distributions; Proof of Gallagher theorem assuming Gallagher lemma |

40 | 4/13 | [No class] |

41 | 4/16 | F.4 Proof of Gallagher lemma average singular series, G.1 Small Gaps between primes: Goldston-Pintz-Yildirim |

42 | 4/17 | G.2 Main estimates, G.3 Goldston-Pintz Yildirim explicit estimates (make-up class) |

43 | 4/18 | [Student projects presentations] Cardinal-Mertens matrices; Cramer V-function generalizations; Pretentious methods in number theory (2 hours) |