In the previous post we defined $latex {p}&fg=000000$-adic numbers. This post will state (mostly without proof) some more surprising results about continuous functions $latex {f \colon \mathbb Z_p \rightarrow \mathbb Q_p}&fg=000000$. Then we give the famous proof of the Skolem-Mahler-Lech theorem using $latex {p}&fg=000000$-adic analysis. 1. Digression on $latex {\mathbb C_p}&fg=000000$ Before I go on,… Continue reading A trailer for p-adic analysis, second half: Mahler coefficients
Tag: analytic number theory
A trailer for p-adic analysis, first half: USA TST 2003
I think this post is more than two years late in coming, but anywhow... This post introduces the $latex {p}&fg=000000$-adic integers $latex {\mathbb Z_p}&fg=000000$, and the $latex {p}&fg=000000$-adic numbers $latex {\mathbb Q_p}&fg=000000$. The one-sentence description is that these are ``integers/rationals carrying full mod $latex {p^e}&fg=000000$ information'' (and only that information). The first four sections will… Continue reading A trailer for p-adic analysis, first half: USA TST 2003
Vinogradov’s Three-Prime Theorem (with Sammy Luo and Ryan Alweiss)
This was my final paper for 18.099, seminar in discrete analysis, jointly with Sammy Luo and Ryan Alweiss. We prove that every sufficiently large odd integer can be written as the sum of three primes, conditioned on a strong form of the prime number theorem. 1. Introduction In this paper, we prove the following result:… Continue reading Vinogradov’s Three-Prime Theorem (with Sammy Luo and Ryan Alweiss)
Linnik’s Theorem for Sato-Tate Laws on CM Elliptic Curves
\title{A Variant of Linnik for Elliptic Curves} \maketitle Here I talk about my first project at the Emory REU. Prerequisites for this post: some familiarity with number fields. 1. Motivation: Arithemtic Progressions Given a property $latex {P}&fg=000000$ about primes, there's two questions we can ask: How many primes $latex {\le x}&fg=000000$ are there with this… Continue reading Linnik’s Theorem for Sato-Tate Laws on CM Elliptic Curves
Proof of Dirichlet’s Theorem on Arithmetic Progressions
In this post I will sketch a proof Dirichlet Theorem's in the following form: Theorem 1 (Dirichlet's Theorem on Arithmetic Progression) Let $latex \displaystyle \psi(x;q,a) = \sum_{\substack{n \le x \\ n \equiv a \mod q}} \Lambda(n). &fg=000000$ Let $latex {N}&fg=000000$ be a positive constant. Then for some constant $latex {C(N) > 0}&fg=000000$ depending on $latex… Continue reading Proof of Dirichlet’s Theorem on Arithmetic Progressions
Power Overwhelming 