## Analytic Number Theory: Diophantine Equations

Math 8440 - Fall 2021

Textbook:      There is no official textbook for the course but I will use

• H. Davenport: Diophantine equations and inequalities
• R. C. Vaughan: The Hardy-Littlewood method

Description:  The aim is to give an introduction to analytic methods to diophantine equations, with emphasis on the Hardy-Littlewood method of exponential sums.
More specifically I plan to discuss the following topics:
• Waring's problem: sums of k-th powers.
• Weyl's inequality and Hua's mean value theorem.
• The singular integrals and the singular series.
• Vinogradov's main conjecture and mean value theorem.
• Efficient congruencing on the model case k=2.
• Efficient congruencing for k=3 and the general case.
• Non-diagonal cubic forms and Davenport's theorem.
• Birch's theorem on general diophantine systems.

I. Waring's problem: the minor and major arcs

We discuss the asymtotics for the number of solutions to Waring's problem dute to Hardy-Littlewood and Hua.
Notes: H. Davenport: Diophantine equations and inequalities

Lecture 1      Lecture 2      Lecture 3      Lecture 4     Lecture 5       Lecture 6        Lecture 7       Lecture 8    Lecture 9    Lecture 10

II. Vinogradov's mean value theorem and efficient congruencing

We discuss Vinogradov's improvement on Waring's problem through his mean value theorem. We also discuss the recent solution to Vingradov's cojecture through the so-called efficient congruencing method.
Notes: Heath-Brown: The cubic case of Vinogradov's Mean Value Theorem: A simplified approach to Wooley's ``efficient congruencing"

Lecture 11      Lecture 12     Lecture 13    Lecture 14    Lecture 15        Lecture 16     Lecture 17      Lecture 18

III. Systems of non-diagonal diophantine equations

We introduce a general form of the circle-method to treat systems of non-diagonal equations, due the Birch and Davenport.
Notes: H. Davenport: Diophantine equations and inequalities

Lecture 19      Lecture 20   Lecture 2        Lecture 22         Lecture 23        Lecture 24     Lecture 25