Primes in arithmetic progressions to spaced moduli. II (Q2922439)

For \(Q>0\), let \(S_Q\) be the set of positive integers contained in the interval \([Q, 2Q]\). Put NEWLINE\[NEWLINE E(x; q,a) = \underset{n\equiv a \pmod{q}}{\sum_{n\leq x}} \Lambda(n) - \frac{x}{\phi(q)}, NEWLINE\]NEWLINE where \(\Lambda\) is von Mangoldt's function. Define the variance NEWLINE\[NEWLINEV(Q,S) = \sum_{q\in S_Q} \underset{(a,q)=1}{\sum_{a=1}^q} E(x; q,a)^2.NEWLINE\]NEWLINE It is proved that for \(S = \{\lfloor nc\rfloor : n \geq 1\}\) where \(c\) is an integer with \(c>1\) we have NEWLINE\[NEWLINEV(Q, S) \ll |S_Q|x^2/Q{\mathcal L}^A,NEWLINE\]NEWLINE for every \(A > 0\), where \({\mathcal L} = \log x\), provided that \(Q<x{\mathcal L}^{-B}\) with \(B = B(c, A) > 0\). Under certain assumptions, a similar type estimate is given for the average maximum error \(M(Q, S)\). The theorems fit well and in some sense extend well-know results from the literature.NEWLINENEWLINEFor Part I see [Acta Arith. 153, No. 2, 133--159 (2012; Zbl 1252.11069)].