On the gap distribution of prime factors

Let { p j stretchy="false">( n stretchy="false">) stretchy="false">} = 1 ω \{p_j(n)\}_{j=1}^{\omega (n)} denote the increasing sequence of distinct prime factors an integer alttext="n"> encoding="application/x-tex">n . For alttext="z greater-than-or-slanted-equals 0"> z ⩾ 0 encoding="application/x-tex">z\geqslant 0 , let alttext="upper G semicolon z G ; encoding="application/x-tex">G(n;z) number those indexes alttext="j"> encoding="application/x-tex">j such that alttext="p plus greater-than exp z"> + &gt; exp ⁡<!-- ⁡ </mml:msup> encoding="application/x-tex">p_{j+1}(n)&gt;p_j(n)^{\operatorname {exp} z} We show uniform convergence, with almost optimal effective estimate speed, distribution on alttext="StartSet colon less-than-or-slanted-equals upper N EndSet"> : ⩽<!-- ⩽ <mml:mi>N encoding="application/x-tex">\{n:1\leqslant n\leqslant N\} to a Gaussian limit law mean alttext="normal e negative log 2 n"> e −<!-- − <mml:mi>log 2 encoding="application/x-tex">\operatorname {e}^{-z}\log _2n variance normal minus encoding="application/x-tex">\{\operatorname {e}^{-z}-2z\operatorname {e}^{-2z}\}\log and we establish asymptotic formula remainder for all centered moments.