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On the divisors of shifted primes

Authors: JEAN MARIE DE KONINCK, IMRE KATAI

Abstract: Let τ(n) stand for the number of positive divisors of n. Given an additive function f and a real number α[0,1), let hn(α):=1τ(n)dn{f(d)}<α1, where {y} stands for the fractional part of y, and consider the discrepancy Δ(n):=sup0α<β<1|hn(β)hn(α)(βα)|. We show that Δ(p+1)0 for almost all primes p if and only if q for every positive integer m, where \|x\| stands for the distance between x and its nearest integer and where the sum runs over all primes q.

Keywords: Sum of divisors function, shifted primes

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