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.

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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.

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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.

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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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Sums of Squares and Density

Classification theorems of Euler, Lagrange, and Legendre describe the sets of integers that can be written as the sum of 2, 3, and 4 squares. In the last two cases, it follows easily that the density of these sets are 5/6 and 1.

The question of density is not so simple in the case of two squares. In this post, we resolve using an unexpected tool — Dirichlet’s theorem on primes in arithmetic progressions.

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Notes on the Chebyshev Theorem

The prime number theorem (PNT) was not proven until 1896, but a weaker form (up to constants) was established decades earlier. The earliest proof was due to Chebyshev in 1852, and his work inspired others to take up the mantle and inch towards what they thought would be a proof of the PNT. Here, we show the strength of the Chebyshev method and ask whether it had the power to prove the PNT after all.

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