# A Proof of Stirling’s Approximation via Contour Integration

Stirling’s approximation gives a useful estimate for large factorials. This post contains a (new?) proof of Stirling’s formula which relies on properties of the Riemann zeta function.

Stirling’s approximation gives a useful estimate for large factorials. This post contains a (new?) proof of Stirling’s formula which relies on properties of the Riemann zeta function.

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.

In this post, we discuss a few ways in which the symmetric and alternating groups can be realized as finite collections of self-maps on the Riemann sphere. We calculate maximal injections of symmetric and alternating groups and discuss when these actions respect a finite invariant set.