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.

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Formal Groups and Where to Find Them

In 1946, S. Bochner published the paper Formal Lie Groups, in which he noted that several classical theorems (due to Sophus Lie) concerning infinitesimal transformations on Lie groups continue to hold when the (convergent) power series locally representing the group law was replaced by a suitable formal analogue. It was not long before this formalism found far-reaching uses in algebraic number theory and algebraic topology.

Unfortunately, few students see more than two or three explicit (i.e. closed form) group laws before stumbling into the deep end of abstract nonsense. In this article, we’ll see in a rigorous sense why this must be the case, providing along the way a complete classification of polynomial and rational formal group laws (over any reduced ring).

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