### MAT 6932

Seminar in Number Theory
on
Irrationality, Diophantine Approximations and Transcendence
MWF 6th period (12:50-1:40pm)
Section 7867 - LIT 239
SPRING 2012

### INSTRUCTOR:

304 Little Hall
392-0281 extn 227

### OFFICE HOURS:

MW 8th period (3:00 - 4:00 pm) in LIT 304 and by appointment.

None

### COURSE DESCRIPTION:

This course is an extension of the one given in Fall 2011.

The study of irrational numbers dates back to Greek antiquity. Yet the subject remains an active area of research today. Although "almost all" real numbers are irrational, it is very difficult to establish the irrationality of a given number. We will systematically study a variety of techniques which will help in confirming the irrationality of different classes of numbers starting with Dirichlet's fundamental criterion for irrationality. Following this, we will discuss irrationality criteria utilizing series and product representations of reals due to Engel, Cantor and Sylvester, regular and general continued fraction expansions, and Farey fractions. We will present various proofs of the irrationality of important numbers such as e and π, and the irrationality of functions like the trigonometric, hyperbolic and the Bessel functions at rational arguments. Next we will take up the study of the closeness of approximation of irrationals by rationals and introduce the concept of irrationality measures. This is the subject of Diophantine approximations. It will be shown that the truncations of the regular continued fraction expansion of real numbers generate the sequence of "best approximations", but it is very difficult to determine the continued fraction expansion of a given irrational. We will develop methods to obtain efficient irrationality measures. This will involve the use of Legendre polynomials and Pade approximations. An important outcome of this approach is the irrationality of the Riemann zeta function at the odd integer 3 - a fact long conjectured but only established in 1978. The study of irrationality measures also explains why Pell's equation such as x2-2y2=1 has infinitely many solutions whereas the Thue equation x3-2y3=1 has only finitely many integer solutions. Thus the subject of Diophantine approximations is closely associated with the theory of Diophantine equations*. As part of our study of Diophantine approximations, we will include a discussion of uniform distribution and of normal numbers - numbers whose decimal digits are statistically uniformly distributed. Finally, we will launch the study of transcendental numbers by showing first that e is transcendental, and more generally that the exponential function takes transcendental values at non-zero algebraic arguments. Since exp(iπ)=-1, this result of Lindemann at the end of the 19-th century implies that π is transcendental and thereby shows the impossibility of "squaring the circle" - one of three problems of Greek antiquity. We also hope to discuss measures of transcendence and the transcendence of some fascinating functions (defined by lacunary series) at algebraic arguments. A host of great problems remain unsolved: Is Euler's constant γ irrational? Are the values of the Riemann zeta function at the odd integers >3 irrational? What is the precise irrationality measure for π? We know that "almost all" numbers are normal to all bases, but we do not know a single example of a number whose digit expansions to all integer bases (not just base 10) are normal.

The course will be completely self contained and can be followed by any of our graduate students and those in allied disciplines as well (such as computer science and statistics). I will use my own detailed notes for the course but will give a number of books as references.

The list of topics given above is vast, and I will cover as many of them as time permits and depending on the interest of the audience. My goal is not to race through the topics, but do a thorough discussion at a pace that is comfortable.

* The topics to be covered in my course augment those discussed by Professor Berkovich in his year long seminar on Diophantine equations during 2010-11.

### TEXT:

No assigned text. I will use my own notes. A number of texts will be given as references.