Numbers Divisible by a Large Shifted Prime and Large Torsion Subgroups of CM Elliptic Curves

Abstract

We sharpen a 1980 theorem of Erdős and Wagstaff on the distribution of positive integers having a large shifted prime divisor. Specifically, we obtain precise estimates for the quantity N (x, y) := #{n ≤ x : − 1 | n for some − 1 > y, prime}, in essentially the full range of x and y. We then present an application to a problem in arithmetic statistics. Let T CM (d) denote the largest order of a torsion subgroup of a CM elliptic curve defined over a degree d number field. Recently, Bourdon, Clark, and Pollack showed that the set of d with T CM (d) > y has upper density tending to 0, as y → ∞. We quantify the rate of decay to 0, proving that the upper and lower densities of this set both have the form (log y) −β+o(1) , where β = 1 − 1+log log 2 log 2 (the Erdős-Ford-Tenenbaum constant).

Denominators of Bernoulli Polynomials

“…McNew, Pollack and Pomerance [17], improving on the previous result of Erdős and Wagstaff [6], have shown that uniformly in 3 ď y ď x,…”

Integers divisible by a large shifted prime

“…uniformly for x > y > 10, and applied this estimate to the study of denominators of Bernoulli numbers. In [7], the following improved estimates were shown. Here log 2 x = log log x, log 3 x = log log log x and δ = 1 − 1+log log 2 log 2 = 0.08607 .…”

“…Research supported in part by NSF grant DMS-1501982. In this paper, we determine the correct order of magnitude for N (x, y) uniformly for all x, y and show an asymptotic for N (x, y) in most of the range (iii) of Theorem A. As in [7], define α implicitly by y = x/ exp{(log x) α }, so that 0 α 1 in the range 1 y x/e. Near the threshold value α = 1 log 4 , define θ by Uniformly in the slightly larger range x/ exp{(log x) 1/ log 4 } y x/2, x 10, we have…”

“…The authors also claim in [Remark 2.10] [7] that they can sharpen (iii) to N (x, y) ≍ x log(x/y) log x by taking more care of the "singular series" factor coming from a sieve estimate. As we shall see below, this is a delicate matter.…”

Triangles with prime hypotenuse

“…Let η = 1 − 1 + log log 2 log 2 ≈ 0.086 be the Erdős-Ford-Tenenbaum constant. This constant is related to the number of distinct products in the multiplication table, and also arises in other contexts, for example, see [3,4,11,12].…”