Brad Rodgers: Best approximation by restricted divisor sums and random matrix integrals

Abstract:

Let \(X\) be large and \(H\) also large but slightly smaller, and consider \(n\) ranging from \(1\) to \(X\). For an arithmetic function \(f(n)\) like the \(k\)-fold divisor function, what is the best mean square approximation of \(f(n)\) by a restricted divisor sum (a function of the sort \(\sum_{d|n, d < H} a_d\))? I hope to explain some of the context around this question and how the answer is connected to random matrix integrals over the unitary group.