In number theory, Dirichlet's theorem (also known as the Dirichlet prime number theorem), states that for any two positive coprime integers $a$ and $d$, there are infinitely many primes of the form $a + nd$, where $n$ is also a positive integer. In this talk, we shall look into an elementary proof for primes in an arithmetic progression. We shall also discuss some further works done in this direction.
How large can the sum of moduli of the terms of a convergent power series be? Harald Bohr addressed this question in 1914 and as a result, he finds a subdisk in the unit disk under which the absolute sum of the Taylor series expansion of the bounded analytic function, bounded by one, is defined in the unit disk is also bounded by one. In this seminar, I will present the proof and generalizations of this remarkable result.
This talk will be elementary. We will discuss the definition and examples of C*-algebras. Then we will see how commutative and non commutative C*-algebras are different by the means of an example. Towards the end, we will discuss how the non commutative torus can be realised as a crossed product C*-algebra and as a braided (twisted) tensor product.
Diophantine approximation is one of the powerful tools to tackle transcendence questions. In this talk I will explain how to use Liouville theorem and Roth's theorem, to prove certain real numbers are transcendental. In particular, we will see that the transcendence of the sums $\sum_{n=1}^\infty 1/2^{n!}$ and $\sum_{n=1}^\infty 1/2^{3^n} $ are immediate consequences of these theorems. I will conclude this talk by presenting my results on the transcendence criterion for Cantor series expansion using the Schmidt Subspace Theorem, which is a higher dimensional generalization of Roth's theorem.