The Euler totient function in short intervals

Abstract

It is known that the two rings Z and Fq[T] have a lot of similar properties. Here Fq[T] denotes the polynomial ring over the ?finite ?field Fq of q elements. In 2014 Keating and Rudnick developed a new technique for calculating the variance of functions in short intervals in Fq[T]. In this thesis we study this technique and apply it to the von Mangoldt function, as Keating and Rudnick did to prove a result analogous to a theorem in Z, due to Goldston and Montgomery. We then apply this technique to the Euler totient function to arrive at some new results concerning its variance in short intervals in Fq[T]. Finally we turn our attention to the Euler totient function in short intervals in Z. Surprisingly, it turns out that the analogue to the statement in Fq[T] does not hold.