# 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.

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.

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.

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.