If a set of positive integers contains no arithmetic progressions, how large can it be? In this post, we study this question in the context of harmonic sums.
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.
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.