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

# Category: 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

## Positive Definite Quadratic Forms

I'm reading through Primes of the Form $latex {x^2+ny^2}&fg=000000$, by David Cox (link; it's good!). Here are the high-level notes I took on the first chapter, which is about the theory of quadratic forms. (Meta point re blog: I'm probably going to start posting more and more of these more high-level notes/sketches on this blog… Continue reading Positive Definite Quadratic Forms

## 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)

## Miller-Rabin (for MIT 18.434)

This is a transcript of a talk I gave as part of MIT's 18.434 class, the ``Seminar in Theoretical Computer Science'' as part of MIT's communication requirement. (Insert snarky comment about MIT's CI-* requirements here.) It probably would have made a nice math circle talk for high schoolers but I felt somewhat awkward having to… Continue reading Miller-Rabin (for MIT 18.434)

## Artin Reciprocity

I will tell you a story about the Reciprocity Law. After my thesis, I had the idea to define $latex {L}&fg=000000$-series for non-abelian extensions. But for them to agree with the $latex {L}&fg=000000$-series for abelian extensions, a certain isomorphism had to be true. I could show it implied all the standard reciprocity laws. So I… Continue reading Artin Reciprocity

## Some Notes on Valuations

There are some notes on valuations from the first lecture of Math 223a at Harvard. 1. Valuations Let $latex {k}&fg=000000$ be a field. Definition 1 A valuation $latex \displaystyle \left\lvert - \right\rvert : k \rightarrow \mathbb R_{\ge 0} &fg=000000$ is a function obeying the axioms $latex {\left\lvert \alpha \right\rvert = 0 \iff \alpha = 0}&fg=000000$.… Continue reading Some Notes on Valuations

## 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

## Zeros and Primes

Prerequisites for this post: previous post, and complex analysis. For this entire post, $latex {s}&fg=000000$ is a complex variable with $latex {s = \sigma + it}&fg=000000$. 1. The $latex {\Gamma}&fg=000000$ function So there's this thing called the Gamma function. Denoted $latex {\Gamma(s)}&fg=000000$, it is defined by $latex \displaystyle \Gamma(s) = \int_0^{\infty} x^{s-1} e^{-x} \; dx… Continue reading Zeros and Primes