## Course Information |
Scheduled
Time |
Room |

Lectures | TTh 10:30 |
Lockett
111 |

Office Hours | T 12:00 |
Lockett
320 |

Textbook |
Hugh Montgomery and Robert Vaughan, Multiplicative Number Theory
I. Classical Theory, Cambridge Studies in Advanced Mathematics 97, Cambridge University Press, 2007. (Available as an e-textbook through LSU libraries) |

- Homework 1 - Due Thursday, Jan. 25
- Homework 2 - Due Thursday, Feb. 1
- Homework 3 - Due Thursday, Feb. 8
- Homework 4 - Due Thursday, Feb. 22
- Homework 5 - Due Thursday, Mar. 1
- Homework 6 - Due Thursday, Mar. 8
- Homework 7 - Due Thursday, Mar. 22
- Homework 8 - Due Thursday, Apr. 12
- Homework 9 - Due Thursday, Apr. 19
- Homework 10 - Due Friday, May 4

- G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford University Press; 6th edition, 2008.
- A. Hildebrand, Introduction
to Analytic Number Theory, Lecture Notes, 2013. link

Papers: (links provided to original sources; subscription or LSU campus-access may be required)

- F. Bayart, The product of
two Dirichlet series, Acta Arithmetica 111 (2004), 141–152. link

- H. Diamond and P. Erdös, On sharp elementary prime number estimates, L'Enseignement Mathématique 26 (1980), 313–321. link
- D. Goldston, Y. Motohashi, J. Pintz, and C. Yıldırım, Small gaps between primes exist, Proceedings of the Japan Academy. Series A. Mathematical Sciences 82 (2006), 61–65. link
- S. Golomb, The lambda
method in prime number theory, Journal of Number Theory
2 (1970), 193–198. link

- A. Granville, Primes in
intervals of bounded length, Bulletin of the American
Mathematical Society 52
(2015), 171–222. link

- G. Hardy and M. Riesz, The general theory of Dirichlet's series, Cambridge Tracts in Mathematics and Mathematical Physics, No. 18, Stechert-Hafner, New York, 1964. link
- D. Hensley and I. Richards, Primes
in intervals, Acta Arithmetica 25 (1973/74), 375–391. link

- H. Maier, Primes in short
intervals, Michigan Mathematical Journal 32 (1985), 221–225. link

- J. Maynard, Small gaps
between primes, Annals of Mathematics 181 (2015), 383–413. link

- J. Maynard, Primes with restricted digits, preprint. arXiv:1604.01041. link
- J. Nicolas and G. Robin, Majorations explicites pour le nombre de diviseurs de N, Canadian Mathematical Bulletin 26 (1983), 485–492. link
- K. Soundararajan, Small
gaps between prime numbers: the work of
Goldston-Pintz-Yıldırım, Bulletin of the American
Mathematical Society 44
(2007), 1–18. link

- Y. Zhang, Bounded gaps between primes, Annals of Mathematics 179 (2014), 1121–1174. link

Links to other resources:

- Online Encyclopedia of Integer Sequences (OEIS)
- MathSciNet
(LSU log-in credentials required)

Date |
Lecture Topics / Reading / Handouts |

Thurs., Jan. 12 |
No class;
begin working on HW 1 |

Tues., Jan. 16 |
Fundamental
Theorem of Arithmetic; Infinitude of primes (P. Clark's
lecture notes);
Prime Number Theorem; Recent history of prime gaps (Y.
Zhang's paper,
J. Maynard's paper,
Polymath 8 website) |

Thurs., Jan. 18 |
Class
canceled due to LSU closure (snow!) |

Tues., Jan. 23 |
Basic
properties of divisor function; Highly Composite Numbers
(OEIS link); Average
value of divisor function (Montgomery-Vaughan [MV] Section
2.1); Discussion of "hyperbola method" |

Thurs., Jan. 25 |
Upper
bounds for divisor function, including Wigert and
Nicolas-Robin; Discussion of repeated logarithms and
"sub-polynomial" functions; Dirichlet's Theorem and the
Divisor Problem (Mathworld page);
Euler's
Summation by Parts (MV Appendix B) |

Tues., Jan. 30 |
Harmonic
series and the Euler-Macheroni Constant (MV Section 1.3);
Cramér's model for the distribution of primes and the
Twin Primes Conjecture (T. Tao's blog);
Bounds for n!, Stirling's formula by Summation by parts (MV
Appendix B; Section 2.2.3 of A. Hildebrand's notes) |

Thurs., Feb. 1 |
Discussion
of Riemann Hypothesis (see A. Granville's survey);
Abel's partial summation, and application to Chebyshev's
equivalent form of PNT; Logarithmic derivative of Riemann
zeta function and von Mangoldt function (MV Section 1.3) |

Tues., Feb. 6 |
Dirichlet series and convolution, including commutative ring structure, Möbius inversion, multiplicative functions |

Thurs., Feb. 8 |
Chebyshev's
Theorem via the Hyperbola Method and Stirling's formula (MV
Section 2.2); Sums of reciprocal primes (MV Section 2.2) |

Thurs., Feb. 15 |
Continuation of sums of reciprocal primes; Upper bound for divisor function (MV Section 2.3; Nicolas and Robin's paper; T. Tao's blog); Introduction to MathSciNet |

Tues., Feb. 20 |
Convergence
properties of Dirichlet series, including abscissae of
convergence and absolute convergence, Stolz angles (MV
Section 1.2; G. Hardy's book;
K. Conrad's notes);
Dirichlet's eta-function (MathWorld) |

Thurs., Feb. 22 |
Uniqueness
of Dirichlet series on a half-plane and polynomial bounds;
Convergence of products of Dirichlet series (MV Section 1.2;
F. Bayart's paper);
Introduction to the Sieve of Eratosthenes (MV Section 3.1) |

Tues., Feb. 27 |
Sieve of
Eratosthenes-Legendre, and failure of "error" term; Bounds
for prime counting function; Basic sieve bound for primes
(MV Section 3.1) |

Thurs., Mar. 1 |
Selberg's
lambda-squared/upper-bound sieve; Applications to upper
bound for primes without Chebyshev's Theorem and absolute
bound for prime "clustering"; Quadratic optimization and
proof of main term (MV Section 3.2) |

Tues., Mar. 6 |
Error
term in Selberg's sieve (MV Section 3.2); Further discussion
of primes in intervals (H. Maier's paper;
D. Hensley and I. Richards' paper) |

Thurs., Mar. 8 |
Selberg's upper-bound Sieve and Twin Primes, main term (MV Section 3.4) |

Tues., Mar. 13 |
Selberg's
upper-bound Sieve for general congruence restrictions; error
term for Twin Primes (MV Section 3.3, J.
Teräväinen's blog) |

Thurs., Mar. 15 |
Introduction
to lower bounds from Selberg's sieve and small prime gaps
(K. Soundararajan's survey);
Admissible k-tuples (Wikipedia page);
Hardy-Littlewood Conjecture |

Tues., Mar. 20 |
Introduction
to group characters (MV Section 4.2; K. Conrad's blurb);
Orthogonality and indicator functions; Multiplicative
Dirichlet characters |

Thurs., Mar. 22 |
L-functions
of Dirichlet characters (MV Section 4.3, A. Sutherland's notes);
Primes in arithmetic progressions; Nonvanishing of
L-functions for non-principal characters |

Tues., Apr. 3 |
Examples of Dirichlet characters, including Jacobi symbols; More details of the proof of Dirichlet's Theorem (MV Section 4.3); Prime Number Theorem in Arithmetic Progressions (MV Section 11.3) |

Thurs., Apr. 5 |
Preparation
for Zhang and Maynard-Tao's work on small prime gaps,
including Dickson tuples; arithmetic/natural density (A.
Granville's survey) |

Tues., Apr. 10 |
Von Mangoldt function and Lambda method for controlling number of prime divisors (S. Golomb's paper); Equidistribution of primes in arithmetic progressions, Bombieri-Vinogradov Theorem (A. Granville's survey) |

Thurs., Apr. 12 |
Goldston-Pintz-Yıldırım (GPY) sieve setup, including Selberg weights, combinatorics of "good" residue classes for an admissible k-tuple (A. Granville's survey, Sections 4.1, 4.2) |

Tues., Apr. 17 |
Diagonalization of quadratic expression via Selberg's construction (A. Granville's survey, Section 4.5); Discussion of generalized divisor functions |

Thurs., Apr. 19 |
Generalized divisor functions and generalization of Hyperbola Method (T. Tao's blog post); Sums of multiplicative functions via Dirichlet Series (A. Granville's survey, Section 4.6), including generalizations of Twin Prime Constant (MV Section 3.4) |

Tues., Apr. 21 |
Discussion of correlations for generalized divisor functions (T. Tao's blog post); First sum in GPY sieve (A. Granville's survey, Section 4.7) ; Approximation of divisor sums by integrals of smooth functions (A. Granville's survey, Section 4.8) |

Thurs., Apr. 26 |
Higher-dimensional GPY sieve (A. Granville's survey, Section 6.1), and diagonalizing the quadratic form (Section 6.2); Maynard's polynomials and successful use of Bombieri-Vinogradov (J. Maynard's paper) |

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