Time: Tuesday 7-9 pm
Location: Math Room 622
Section Leader: Chung-Hang (Kevin) Kwan
Many good stories start with "once upon a time" and this include number theory! Number theory has been a source of fascination for millennia and is described by many as the "queen of mathematics" for its beauty in the purest form. Classically, it refers to the study of the interesting qualitative properties of numbers, but in this seminar the audience will be introduced to the global and quantitative perspectives of number theory.
To illustrate this, let's consider the sequence of prime numbers: 2,3,5,7,11, ... , 2^13466917-1, 143332^786432-143332^393216+1, ... The sequence of prime numbers seem to have no pattern and prime numbers become rare when they get large.
In terms of qualitative perspective, one hopes for a prime-producing formula. There has been a lot of failed attempts (say Euler & Fermat). Finally, mathematicians did found one in 1976 but it is extremely complicated! All prime numbers can be captured by a system of 14 equations in 26 variables (just enough to be expressed in terms of English alphabets!), which can be be seen in this Wiki page . Of course, such formula has no practical use in the investigation of prime numbers.
In terms of quantitative perspective, we may ask for the rarity/ abundance of prime numbers. Essentially, we want to count how many primes are there (say from 1 to x). Amazingly, there is a very simple formula for this --- there are roughly (x/log x)-many primes from 1 to x, as x goes to infinity. This is known as the Prime Number Theorem and undoubtedly it is one of the major triumph in the modern number theory! As a side note, it is quite remarkable that Gauss conjected this formula simply by staring at the table for prime numbers (probably for a long time)!
This will be a student-oriented seminar. Each student enrolled in this section is expected to read through some short mathematical articles which are relevant to the scopes of this seminar, and then give a talk, share your passion with fellow classmates. There are two possibilities for choosing a topic:
Both options are equally good. In any case, feel free to share your thoughts, interests and backgrounds with me before you choose a topic. I am sure we can come up with something fun together! Students with similar interests may form a group to study a single topic (with a couple of sub-topics so that every member has different things to present on). It is always a good idea to discuss with your classmates when you have any problem. It is beneficial to learn the perspectives from others that are different from yours. The maximum number of students allowed in a group depends on the chosen topic. (In any case no more than 4.)
I hope this seminar will provide students the experience of learning mathematics actively, independently, and being able to explain the mathematical concepts he/she has learnt to others effectively. Another important goal is to follow the footsteps of mathematicians in the past and appreciate their brilliant yet simple ideas.
Please email me if you would like to join this seminar.
Here, we DO NOT aim at proving the full Prime Number Theorem as its proof is quite technical (arguably it is a deep theorem) and we do not want to overlap with our Analytic Number Theory class. (But I strongly recommend you to take if you find this topic interesting during the seminar, and after you have met its pre-requisite!)
The Prime Number Theorem was proven in the year 1896 by Hadamard and de la vallee Poussin. But we shall rewind by 50 years to see how Chebyshev deduced many strikingly accurate information about the distribution of prime numbers, skillfully and elementarily.
References:We would like to gain more statistical understanding for prime numbers apart from its count. For example, one may ask how small and how large the gaps of prime numbers can be, how prime numbers correlate with one another, how the prime-counting function oscillate. Admittedly these are very hard problems till this date, but there has been remarkable progress in recent years, see this Quanta Article and this New York Times Article .
We will not be able to touch upon these amazing advances as they are very deep. Nevertheless, it is possible for us to make a good guess about what should be the correct answers to these questions based on probabilistic model, which simply builds upon the Prime Number Theorem.
References:To those interested in some concrete results about prime gaps, I suggest the results of Erdos and Rankin back in 1930s, which are short and not too hard to understand.
References:A lot of problems about prime numbers (and beyond) can be formalized in the framework of Sieve Theory. The principle of sieve theory is very simple. Many of you may remember the Sieve of Eratosthenes dated back to ancient Greek. The goal of such sieve is to find out the prime numbers in a range. We cross out the multiples of 2 except 2, the multiples of 3 except 3, the multiples of 5 except 5, so on and so forth. What remain will be prime numbers.
Such idea was left dormant by 2000 years until the early 20th century, when V. Brun introduced his novel idea in the study of twin primes (i.e., pairs of prime numbers differ by 2). He obtained two revolutionary results (in fact, the very first progress towards the Twin Prime Conjecture, i.e., there are infinitely many twin primes.): (1). the reciprocal sum of all pairs of twin primes is convergent! (the value of such series is called Brun's constant); (2) there are infinitely many pairs of numbers differ by 2 and each has at most 9 prime factors.
Sieve methods remain one of the most powerful methods in studying problems about prime numbers and there has been very significant development in the past century. This part of the seminar merely serves as a taster of such a huge and fascinating area. I suggest the following sub-topics that can be handled pretty elementarily:
In classical number theory, one asks for the possibility of expressing every (or sufficiently large) positive integer as a finite sum of certain interesting sequence. One of the first and the most celebrated results in this direction was due to Lagrange in 1770, which asserts that every natural number is a sum of four perfect squares (e.g., 13=0^2+0^2+2^2+3^2, 31=1^2+1^2+2^2+5^2). We may then say that the set of perfect squares is a basis of order 4. This is also one of my favourite theorem I have learnt in my first course of number theory.
Similarly, one can ask if the set of perfect cubes, biquadratics, quintics etc. (known as Waring's problem), or even the set of primes (known as Goldbach's problem), serve as bases of finite order. Collectively, these are called basis problems.
There has been significant progress in this area (known as additive number theory) using more analytic, sophisticated machinery, such as the circle method and the sieve methods, and one is able to obtain rather precise results. For example, it is now known that every odd number not less than 7 is a sum of three prime numbers. Thus, primes serves as a basis of order 4. It is conjected that we can do a bit better: the set of primes is in fact a basis of order 3. Moreover, every natural number is a sum of 9, 19, 37 perfect cubes, biquadratics, quintics respectively. If one only requires sufficiently large natural number to be expressible as perfect cubes, biquadratics, quintics, one can significantly reduce the numbers to 7, 16, 17 respectively.
However, our goal here is much more modest and we shall stick with the elementary methods only. Of course, the trade-off is that the results obtained in this manner are weaker, but stunningly not by too much! The key is to define the correct notion of the densities of sequences, which is known as the Schnirelmann density, and study its properties. Then we will readily be able to prove that
Another problem about basis is known as Sidon's Thin Basis Problem. We ask for the existence of "thin" (or "efficient") basis of order k for any positive integers k. We say a basis is thin if the number of ways to represent any natural number n by that basis is small (of the size log n). This turns out to exist, but the surprising part is that we prove its existence without writing one down! This is one of the advantages of using probabilistic methods.
References:We will motivate this topic by some (seemingly) simple problems:
Let p be a prime and a be an integer not divisible by p. How small can positive integers m,n <= (p-1) be, such that (mn-a) is divisible by p?
Remarkably, this is still an open problem! We shall see how to prove non-trivial results through the bounding of a special type of exponential sum.
Lattice Point Counting. Draw a circle with radius R on a coordinate plane. How many points with integral coordinates are lying inside such circle?
A first guess is of course the area of the circle (pi*R^2)! This is a good approximation but we want to do better. However, due to the curvature of circles, there is considerabe fluctuation in the count as you expand the size of the circle and this makes the problem hard. Indeed, this is an open problem for hundreds of years and is known as Gauss' Circle Problem. We shall see how to make partial progress towards this problem with the knowledge of exponential sums/integrals and what is the best possible answer one can get.
The schedule is tentative and subject to change as the seminar moves along.
| Date | Speaker | Topic | References | |
| Lecture 1 | 9/10 | Kevin | Introduction | |
| Lecture 2 | 9/17 | Kevin | Introduction | |
| Lecture 3 | 9/24 | Amara | The Primes that Euclid Forgot | |
| Lecture 4 | 10/1 | Maya & Cecilia | Randomness, Benford's Law & Diophantine Approximation | |
| Lecture 5 | 10/8 | Catherine & Maggie | The Multiplication Table and the American Flag | |
| Lecture 6 | 10/15 | Gia & Sung-Jun | Ramanujan and partition function | |
| Lecture 7 | 10/22 | Pallavi & Jarek | Probabilistic Methods in Constructive Number Theory | |
| Lecture 8 | 10/29 | Helen, Mariya | Monte Carlo algorithm and primality testing, Euler's polynomial and unique factorization | |
| Lecture 9 | 11/12 | Alec & Joon | Turan's sieve & Counting Irreducible Polynomials | |
| Lecture 10 | 11/19 | Rohan & Abdoulaye | Lattice Point Counting Inside Circles | |
| Lecture 11 | 11/26 | Kexin, Akiva | Square Sieve, Anatomy of Middle Binomial Coefficients | |
| Lecture 12 | 12/3 | Ben, William | TBC |
Last updated: September 2, 2019.
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