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)∑d∣n{f(d)}<α1, where {y} stands for the fractional part of y, and consider the discrepancy Δ(n):=sup0≤α<β<1|hn(β)−hn(α)−(β−α)|. 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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