Counting irreducible binomials over finite fields (Q2634386)

For a positive integer \(t\) and prime power \(q\), let \(N_t(q)\) denote the number of irreducible \(x^t-a\in \mathbb F_q[x]\). Let \(\text{rad}(t)\) denote the square-free part of \(t\). It is well-known that \(x^t-a\) is irreducible if and only if (1) \(\text{rad}(t)\mid \text{ord}_q(a)\), (2) \((t, (q-1)/\text{ord}_q(a))=1\) and (3) if \(4\mid t\) then \(q\equiv 1\pmod4\). Using several classical and modern results of analytic number theory, the authors prove a handful of estimates on the average value of \(N_t(q)\). A sample result: There exists an absolute constant \(L>0\) such that uniformly over real \(Q\) and positive integers \(t\) with \(Q\geq t^8\) we have: \[ \sum_{q\leq Q} N_t(q) \geq L\frac{Q^2(\log \text{rad} (t))^2}{\varphi(\text{rad}(t))^2(\log Q)^2}. \] The proof of this result depends on a quantitative version of the Linnik Theorem due to \textit{J. Maynard} [Acta Arith. 157, No. 3, 249--296 (2013; Zbl 1321.11099)].