Sur la répartition du noyau d’un entier

We investigate the asymptotic behaviour of the number N(x,y)N(x,y) of those integers n⩽xn⩽x with squarefree kernel k(n)⩽yk(n)⩽y. Using a double saddle-point method, we obtain an asymptotic formula with remainder that holds, for any given ε>0ε>0, uniformly in the domain y>e(log2x)3+ε. This depends on the saddle-point parameters, defined as the solutions of a transcendental system and for which explicit estimates are provided. This result is in turn exploited to obtaining various explicit estimates for N(x,y)N(x,y). For instance, writing F(t)≔6π2∑m⩾1min(1,et/m)∏p|m(p+1)(t⩾0), and Yx≔e142logx(log2x)3/2, Mx≔2logxlog2xlog3x where logklogk stands for the kk-th iterated logarithm, we show that N(x,y)∼yF(v)⇔y>Yxe−3Mx/8eψxlogxlog2x(v≔log(x/y)) for some function ψx→∞ψx→∞. We also define an explicit function K=K(x,y)K=K(x,y) such that, as x→∞x→∞, N(x,y)∼yF(v)e−{1+o(1)}K(x⩾y⩾2). More precise formulae describe quantitatively the transition phase between the two behaviours N(x,y)∼yF(v)  and  N(x,y)∼yF(v)o(1)(x→∞), the latter occurring if and only if logy=o(logxlog2x).Other consequences of the main formulae are : (i) the exact determination of the size of the factor lost by application of a Rankin type bound; (ii) the derivation of precise formulae for the local behaviour of N(x,y)N(x,y) with respect to both variables, e.g.(∀b>1)N(x,2y)∼2bN(x,y)⇔logy=(logx)1/(b+1)+o(1); (iii) the complete solution of a problem of Erdős and de Bruijn related to the sum K(x)≔∑n⩽x1k(n); and (iv) a new, refined, and heuristically optimal, form of the abcabc conjecture. This last application is detailed in a forthcoming work in collaboration with C.L. Stewart.