Integrating Secant
The integral formula for secant is often introduced in a very artificial way. In this post, I look at the history of this integral and give several derivations.
Large Prime Factors of a Quadratic Polynomial
One of Landau’s four problems from 1912 concerns the infinitude of primes in the values of a certain quadratic polynomial.
In this post, we show that the largest prime factors of the values of this polynomial are “relatively large” infinitely often.
Dirichlet’s Theorem and Sieves
Dirichlet’s Theorem on the infinitude of primes in arithmetic progressions relies on the non-vanishing of non-trivial Dirichlet characters at 1.
In this post, I’ll show how this reduction can be introduced in an intuitive way via sieve theory. If we actually sieve, we obtain estimates for the number of integers whose prime factors lie in given congruence classes.
Sums of Small Divisors
Which integers are a multiple of the sum of their “small” divisors? In the post, we study whether this set should be finite or not.
Our partial solution relates this problem to many outstanding conjectures in number theory about the distribution of prime numbers, such as the twin prime conjecture and the infinitude of Mersenne primes.
The Philosophy of Square-Root Cancellation
In this note, I’ll discuss why square-root cancellation is so typical in problems in number theory and give a quick survey of important sums known or widely conjectured to satisfy bounds of this form.
Two Classic Problems in Point-Counting
This post discusses two classic problems in analytic number theory: the Gauss circle problem and the Dirichlet divisor problem.
These problems are known to be related at a deep level, a fact which is often missed at first glance because the obvious/early attacks on them look quite different.
In this post, I compare these “trivial” estimates, and show how Gauss’ estimate can be realized using a few different techniques.
Notes on Riemann Sums
For many calculus students, Riemann sums are those annoying things that show up in the derivation of the arc length formula.
In truth, these handy sums have done so much more. In this post, I’ll give some examples of Riemann sums dating from before the birth of calculus and some applications of Riemann sums that are still used today.
Counting Matrices of Small Trace
One of the ingredients in a paper that Bram Petri and I submitted in 2016 was a count of integer matrices of determinant 1 with non-negative entries and bounded trace.
Our paper only required an upper bound, but as a number theorist I couldn’t resist the temptation of describing the asymptotics of this function more precisely. In this post we explore do just that, exploring Dirchlet’s hyperbola method along the way.
Lattice Points in High-Dimensional Spheres
The Gauss Circle Problem is a classic open problem in number theory concerning the number of lattice points contained in a large circle.
Optimal error bounds are known in these approximations in a generalization of the Gauss Circle Problem to spheres in dimensions four and above.
In this post, I’ll give a purely analytic proof of this result for even dimensions greater than four, and explain why the method fails in the other cases.
Prime-Generating Polynomials
Just how good are polynomials at producing primes? Do there exist polynomials that produce primes for arbitrarily many consecutive inputs?
In this post, I’ll give a brief overview of what we expect to be able to prove, and show how interpolating polynomials can produce record-breaking prime-generators. (And then break a record, because why not?)
a. w. walker 