# 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.

# 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.

# Maximal Products of a Given Sum

Back in high school, I came across the following contest problem – “What is the largest product of a set of positive integers totaling 20?”

It’s a fun problem, so don’t rush past the spoiler tags too fast. In this post, we’ll give the solution to this problem and discuss a “continuous” version of this question. Namely, what happens when we’re allowed to include real numbers in our product?