In this post we'll make sense of a holomorphic square root and logarithm. Wrote this up because I was surprised how hard it was to find a decent complete explanation. Let $latex {f : U \rightarrow \mathbb C}&fg=000000$ be a holomorphic function. A holomorphic $latex {n}&fg=000000$th root of $latex {f}&fg=000000$ is a function $latex {g… Continue reading Holomorphic Logarithms and Roots

# Tag: complex analysis

## Formal vs Functional Series (OR: Generating Function Voodoo Magic)

Epistemic status: highly dubious. I found almost no literature doing anything quite like what follows, which unsettles me because it makes it likely that I'm overcomplicating things significantly. 1. Synopsis Recently I was working on an elegant problem which was the original problem 6 for the 2015 International Math Olympiad, which reads as follows: Problem… Continue reading Formal vs Functional Series (OR: Generating Function Voodoo Magic)

## 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