From highly composite numbers to transcendental number theory

Math 249 A Fall 2010 : Transcendental Number Theory

α is algebraic if there exists p ∈ Z[x], p 6= 0 with p(α) = 0, otherwise α is called transcendental . Cantor: Algebraic numbers are countable, so transcendental numbers exist, and are a measure 1 set in [0, 1], but it is hard to prove transcendence for any particular number. Examples of (proported) transcendental numbers: e, π, γ, e, √ 2 √ 2 , ζ(3), ζ(5) . . . Know: e, π, e, √ 2 √ 2 are transce...

Galois Theory, Motives and Transcendental Numbers

From its early beginnings up to nowadays, algebraic number theory has evolved in symbiosis with Galois theory: indeed, one could hold that it consists in the very study of the absolute Galois group of the field of rational numbers. Nothing like that can be said of transcendental number theory. Nevertheless, couldn’t one associate conjugates and a Galois group to transcendental numbers such as π...

Transcendental Numbers and Zeta Functions

The concept of “number” has formed the basis of civilzation since time immemorial. Looking back from our vantage point of the digital age, we can agree with Pythagoras that “all is number”. The study of numbers and their properties is the mathematical equivalent of the study of atoms and their structure. It is in fact more than that. The famous physicist and Nobel Laureate Eugene Wigner spoke o...

Algebraic and Transcendental Numbers from an Invitation to Modern Number Theory

The role of complex conjugation in transcendental number theory

In his two well known 1968 papers “Contributions to the theory of transcendental numbers”, K. Ramachandra proved several results showing that, in certain explicit sets {x1, . . . , xn} of complex numbers, one element at least is transcendental. In specific cases the number n of elements in the set was 2 and the two numbers x1, x2 were both real. He then noticed that the conclusion is equivalent...