On sequences involving primes (Q2893515)

Let \(p_n\) be the \(n\)th prime. The main result of the paper is the following extension of distribution of a sequence \(f(n)\bmod 1\) described in \textit{O. Strauch} and \textit{O. Blažeková} [Unif. Distrib. Theory 6, No. 2, 45--63 (2006; Zbl 1153.11038)] replacing \(n\) by \(p_n\): (I) Let the real-valued function \(f(x)\) be strictly increasing for \(x\geq 1\) and let \(f^{-1}(x)\) be the inverse function of \(f(x)\). Suppose thatNEWLINENEWLINE\(\bullet \lim _{k\to \infty}f^{-1}(k+1)-f^{-1}(k)=\infty \),NEWLINENEWLINE\(\bullet\lim _{k\to \infty}\frac {f^{-1}(k+w_k)}{f^{-1}(k)}=\psi (u)\) for every sequence \(w_k\in [0,1]\) for which \(\lim _{k\to \infty}w_k=u\), where this limit defines the function \(\psi (u)\) on \([0,1]\),NEWLINENEWLINE\(\bullet \psi (1)>1\). Then the sequence \(f(p_n)\bmod 1\), \(n=1,2,\dots \), has distribution functions of the form NEWLINE\[NEWLINE g_{u}(x)= \frac {1}{\psi (u)}\cdot \frac {\psi (x)-1}{\psi (1)-1}+\frac {\min (\psi (x),\psi (u))-1}{\psi (u)}, NEWLINE\]NEWLINE where \(u\) runs \([0,1]\).NEWLINENEWLINEResult (I) implies that \(\log p_n\bmod 1\) and the sequences \(\log (p_n\log ^{(i)}p_n)\mod 1\), \(i=1,2,\ldots \) have the same distribution function as the sequence \(\log n\mod 1\). In this paper the author also proved: (II) Every uniformly distributed (u.d.) sequence \(x_n\mod 1\) is statistically independent of \(\log p_n\mod 1\), i.e., \(x_n\mod 1\) and \((x_n+\log p_n)\mod 1\) are u.d. simultaneously. This implies that, for every irrational \(\theta \) the sequence \(p_n\theta +\log p_n\) is u.d. mod 1. (II) Every u.d. sequence \(x_n\mod 1\) is statistically independent with \(\frac {p_n}{n}\mod 1\). Thus \(p_n\theta +\frac {p_n}{n}\) is u.d. mod 1. In proof of (I) the author used the prime number theorem in the form \(\pi (x)=\frac {x}{\log x-1}+O\big (\frac {x}{(\log x)^3}\big)\) and an old result of \textit{M. Cipolla} (1902) \(p_n=n\log n+n(\log \log n-1)+ o\big (\frac {n\log \log n}{\log n}\big)\).