2012/06/11 by Paul Pollack, Lola Thompson, Pollack, Paul +1
Mathematics · #11N25 #11N37 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11N25 #msc:11N37
paper · pdf · doi:10.48550/arxiv.1206.2084
arxiv created 2012/06/11 · arxiv updated 2012/06/12
Fix a field F. In this paper, we study the sets \DF(n) ⊂ [0,n] defined by [\DF(n):= 0 ≤ m ≤ n: Tn-1has a divisor of degree m in F[T].] When \DF(n) consists of all integers m with 0 ≤ m ≤ n, so that Tn-1 has a divisor of every degree, we call n an F-practical number. The terminology here is suggested by an analogy with the practical numbers of Srinivasan, which are numbers n for which every integer 0 ≤ m ≤ σ(n) can be written as a sum of distinct divisors of n. Our first theorem states that, for any number field F and any x ≥ 2, [#F-practical n≤ x \asympF \fracxlogx;] this extends work of the second author, who obtained this estimate when F=\Q. Suppose now that x ≥ 3, and let m be a natural number in [3,x]. We ask: For how many n ≤ x does m belong to \DF(n)? We prove upper bounds in this problem for both F=\Q and F=\Fp (with p prime), the latter conditional on the Generalized Riemann Hypothesis. In both cases, we find that the number of such n ≤ x is ≪F x/(logm)2/35, uniformly in m.