Welcome to Absolutely Sure! This is my new blog, and a companion to the already-existing existing Almost Sure, which is a `random mathematical blog’ concentrating on probability theory and stochastic calculus. In contrast, Absolutely Sure will focus on pure mathematics and,more generally, any mathematical content which does not fit into the category of probability theory.
While the potential scope of this new blog is quite wide, encapsulating all of mathematics, there are some subjects which I plan to kick off with.
- The Riemann Zeta function, Dirichlet Series, and L-series.
- The prime number theorem and Dirichlet’s theorem on primes in an arithmetic progression.
- The Riemann Hypothesis.
- p-adic numbers, Valuation Theory, and Adelic numbers.
In particular, I would like to look at approaches to the Riemann hypothesis, which is one of the great unsolved problems of mathematics. It has been solved in the case of zeta functions over function fields, or algebraic varieties over finite fields. We will look at some of the known proofs for function fields, which many researchers have tried to extend to the number field case. In particular, Enrico Bombieri’s proof for the function field case can be understood with just a knowledge of valuation theory, whereas other methods require some algebraic geometry. Although the Riemann hypothesis is unsolved, there are some partial results which we will look at — such as zero-free regions of the zeta function in the critical strip and the proof that a positive proportion of the zeros do lie on on the critical line.
Besides the ideas suggested above, there are many different topics which could be covered here.