On the divisors of shifted primes

Authors: JEAN MARIE DE KONINCK, IMRE KATAI

Abstract: Let $\tau(n)$ stand for the number of positive divisors of $n$. Given an additive function $f$ and a real number $\alpha\in [0,1)$, let $\displaystyle{h_n(\alpha):= \frac 1{\tau(n)} \sum_{d\mid n \atop \{f(d)\} <\alpha}1}$, where $\{y\}$ stands for the fractional part of $y$, and consider the discrepancy $\Delta(n):= \sup_{0\le \alpha <\beta < 1} |h_n(\beta)-h_n(\alpha) -(\beta-\alpha)|$. We show that $\Delta(p+1) \to 0$ for almost all primes $p$ if and only if $\displaystyle{\sum_{q\in \wp} \frac{\| mf(q) \|^2}q =\infty }$ 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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