circle method and selberg sieve
[02-28-24]
Two of the things that sound the coolest in Analytic Number theory are the circle method and sieves, e.g., selberg sieve and large sieve. I'm slowly learning about these. Here are some preliminary things I've learned.
the circle method: warring's problem
[04-07-24]
The circle method is a quite cool technique in analytic NT. Here is an exposition of the circle method for solving the Warring problem. Much of the analysis follows closely to Nathanson's book on classical additive bases, although in a few places I have tried to give simpler proofs of some of the statements, or otherwise deviated from his presentation to hopefully add clarity and motivation.
number theory classical additive bases: chapter 4
[03-06-24]
A major topic in additive number theory is the following question: given a set $A\subseteq \N$, can every sufficiently large positive integer be expressed as the sum of a bounded number of elements of $A$? If so, in how many ways can number $n$ be represented. We say that the set $A$ is an **additive basis** of order $h$ if every positive integer can be written as the sum of at most $h$ elements from $A$.
Basic properties of primes based on residue mod 4
[07-12-23]
Again #probfromthebook, this time discussing the classes of primes congruent to $1,3 \mod 4$.
overlap of simply periodic binary functions
[05-29-23]
Let $f_a(x)$ denote a binary string consisting of $a$ $0$'s followed by $a$ $1$'s and then repeating these $2a$ symbols. We can equivalently describe $f_a$ as $f_a(x)= \floor{x/a}\bmod 2$. We study $\sum_{x\in [ab]} f_a(x)\oplus f_b(x)$, in particular bounding the difference of the sum from $\frac{ab}{2}$.
Dirichlet's Theorem
[04-23-24]
In this post I will give a proof of Dirichlet's Theorem: in any (reasonable) infinite arithmetic progression there are infinitely many primes. Actually we will probably get a nice density statement that is stronger than this.
Analytic Number Theory - Part 1
[02-22-24]
I've found a cool set of lecture notes from Harvard that talks about analytic number theory. In this post I give some of the basic ideas presented in the first couple notes.
Farey Numbers
[07-23-23]
The $k$-Farey numbers are reduced fractions of denominator at most $k$. They have several very interesting properties and are super useful. We investigate some of these properties.
infinitely many primes one more than a multiple of p
[07-10-23]
Again #probfromthebook, this time discussing the infinitude of some classes of primes.
modular forms
[04-09-24]
Modular forms sound freaking fancy. This post defines the modular group and describes its action on the upper half complex plane. It's actually a really simple group, it's generated by two really simple transformations. It makes some really nice pictures. Anyways this post should be really chill it's just some basic geometric facts. I think it's kind of cute though.
