Consecutive neighbour spacings between the prime divisors of an integer

Abstract

. Writing p 1 ( n ) < · · · < p r ( n ) for the distinct prime divisors of a given integer n ≥ 2 and letting, for a fixed λ ∈ (0 , 1] , U λ ( n ) := # { j ∈ { 1 , . . . , r − 1 } : log p j ( n ) / log p j +1 ( n ) < λ } , we recently proved that U λ ( n ) /r ∼ λ for almost all integers n ≥ 2 . Now, given λ ∈ (0 , 1) and p ∈ ℘ , the set of prime numbers, let B λ ( p ) := { q ∈ ℘ : λ < log q log p < 1 /λ } and consider the arithmetic function u λ ( n ) := # { p | n : ( n/p, B λ ( p )) = 1 } . Here, we prove that (cid:80) n ≤ x ( u λ ( n ) − λ 2 log log n ) 2 = ( C + o (1)) x log log x as x → ∞ , where C is a positive constant which depends only on λ , and thereafter we consider the case of shifted primes. Finally, we study a new function V ( n ) which counts the number of divisors of n with large neighbour spacings and establish the mean value of V ( n ) and of V 2 ( n ) .