On the degrees of divisors of \(T^{n}-1\) (Q374052)

Fix a finite field \({\mathbb F}\). In the paper under review, the authors look at the sets NEWLINE\[NEWLINE {\mathcal D}_{\mathbb F}(n)=\{0\leq m\leq n: T^n-1~{\text{ has a divisor of degree }}~m~{\text{ in }}~{\mathbb F}[T]\}. NEWLINE\]NEWLINE When \({\mathcal D}_{\mathbb F}\) consists of all integers \([0,n]\), then \(n\) is called an \textit{\({\mathbb F}\)-practical number} in analogy with the notion of practical numbers which is similar to the present one in case \({\mathbb F}\) is replaced by \({\mathbb Q}\). The authors prove several theorems about the sets \({\mathcal D}_{\mathbb F}(n)\). For example, they show that the counting function up to \(x\) of the set of \({\mathbb F}\)-practical numbers has order of magnitude \(x/\log x\) as \(x\) tends to infinity. Assume now that \(m\geq 2\) is given. The authors ask what is the counting function of the set of \(n\leq x\) such that \(m\in {\mathcal D}_{\mathbb F}(n)\). They prove that it is \(O(x/(\log m)^{\delta})\), where we can take any \(\delta<1-(1+\log\log 2)/\log 2\) when \({\mathbb F}\) is replaced by \({\mathbb Q}\), and we can take \(\delta=2/35\) when \({\mathbb F}={\mathbb F}_p\) is the finite field with \(p\) elements this last result being conditional under the GRH. The results in this paper are analogs of results obtained by the second author in her Ph.D. dissertation when \({\mathbb F}\) is replaced by \({\mathbb Q}\). The proofs use results of \textit{K. Ford} [Ann. Math. (2), 153, No. 2, 367--433 (2008; Zbl 1181.11058)] and \textit{G. Tenenbaum} [Ann. Sci. Éc. Norm. Supér. (4) 19, No. 1, 1--30 (1986; Zbl 0599.10037)] concerning the distribution of integers \(n\leq x\) with a divisor in an interval \((y,z]\), results concerning the distribution of the multiplicative order \(\ell(p)\) modulo \(p\) of a fixed integer \(a\neq 0,\pm 1\) for varying prime \(p\) due to \textit{P. Kurlberg} and \textit{C. Pomerance} [Algebra Number Theory 7, No. 4, 981--999 (2013; Zbl 1282.11131)], and results concerning the distribution of positive integers \(n\) with small values of the Carmichael \(\lambda\)-function due to \textit{J. B. Friedlander, C. Pomerance} and \textit{I. E. Shparlinski} [Math. Comput. 70, 1591--1605 (2001; Zbl 1029.11043)].