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Description

The correlations between $d(n)$ and $d(n+h)$, where $d(n)$ is the divisor-counting function and $h$ is a possibly varying non-zero integer are a classical topic in analytic number theory, going back to Ingham. It is intimately related to the fourth moment of the Riemann zeta function. After a quick review of the history of this problem, we will discuss two recent variants that arose in our work. In the first variant, we will replace the integers with shifted integers $n+\alpha$, where $\alpha$ is an irrational number. This arose in work joint with Winston Heap on the fourth moment of the Hurwitz zeta function and has connections to Diophantine approximation. In the second variant, we will replace the integers with the ring of polynomials over a finite field. This was investigated in work joint with Alexandra Florea, Matilde Lalín, and Amita Malik, where we extended the range of uniformity in $h$ for which an asymptotic formula is available in this setting, building on earlier work of Conrey--Florea, Gorodestky, Woo and Yiasemides. The main new input here is a Voronoi summation formula for the divisor function, which appears to be novel in this setting.  

https://ucdavis.zoom.us/j/96750070038?pwd=0IWRYx8kJRXkfbc82eaFOo5HBhr8Ei.1