Some almost all results concerning the Pjateckii-Shapiro prime numbers (Q1814860)

Let \(r>0\) be a fixed number. Piatetski-Shapiro primes are the primes of the form \(p=[n^r]\) for some positive integer \(n\). It is known that \(\pi_r(x) =|\{p\leq x: p= [n^r]\} |\sim x^\gamma/ \log x\) where \(\gamma= 1/r>13/15\). The authors study the distribution of Piatetski-Shapiro primes in arithmetic progressions. Using the Bombieri-Vinogradov theorem and exponential sums estimates, they establish the following ``almost all'' results: Theorem 1. Let \(A>0\) and \(\varepsilon>0\) be any constants and let \(\varepsilon\) be sufficiently small. Then for all \(r\in (1,2-4 \varepsilon)\) we have \[ \sum_{q\leq Q} \max_{(a,q)=1} \Bigl|\sum_{{p=[n^r] \atop p\equiv a(q)}}1- {1\over\varphi (q)} \pi_r(x) \Bigl |\ll_{r,A,\varepsilon} x^\gamma/ \log^Ax, \] where \(Q=x^{\gamma-1/2 -\varepsilon}\). Theorem 2. Let \(a\neq 0\) be a fixed integer, \(A>0\), \(\varepsilon>0\) \((\varepsilon\) is sufficiently small). Then for almost all \(r\in (1,2- \varepsilon)\) we have \[ \sum_{{q \leq Q\atop (a,q)=1}} \Bigl |\sum_{{p=[n^r] \atop p\equiv a(q)}}1-{1\over \varphi (q)} \pi_r(x) \Bigr|\ll_{r,A,\varepsilon} x^\gamma/ \log^Ax, \] where \(Q=x^{c(r)- \varepsilon}\) and \[ c(r)= \min\{1/2,2/r-1\}=\begin{cases} 1/2 \quad & \text{if }1<r \leq 4/3 \\ 2/r-1 \quad & \text{if } 4/3 <r<2 \end{cases}. \]