MATH 7230 - Spring 2018

Analytic Number Theory


This course is an introduction to analytic number theory, the goal of which is to determine precise estimates for arithmetic functions and primes. We will cover classical and modern results, with a particular focus on recent advances in sieve methods (with applications to prime gaps). See the Syllabus and detailed Lecture Schedule for more details.

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)



Problem Sets



Other references

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

Links to other resources:



Lecture Schedule


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

Thurs., Feb. 22

Thurs., Mar. 1

Tues., Mar. 6

Thurs., Mar. 8

Tues., Mar. 13

Thurs., Mar. 15

Tues., Mar. 20

Thurs., Mar. 22

Tues., Apr. 3

Thurs., Apr. 5

Tues., Apr. 10

Thurs., Apr. 12

Tues., Apr. 17

Thurs., Apr. 19

Tues., Apr. 21

Thurs., Apr. 26


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