On a variance associated with the distribution of primes in arithmetic progressions. (Q2766399)

As usual, denote \(\psi(x,q,a)=\smash{\sum_ {n \leq x; n \equiv a \bmod q}}\Lambda(n)\), where \(\Lambda\) is von Mangoldt's function, and write \(V(x,q) = \sum_ {1 \leq a \leq q; (a,q)=1}{| \psi(x,q,a)-x/\phi(q)| }^ 2\). Then well-known theorems due to M. B. Barban, H. Davenport and H. Halberstam, C. Hooley, and others, study the quantity \(\sum_ {q \leq Q}V(x,q)\). For individual \(q\), \textit{C. Hooley} [J. Lond. Math. Soc. (2) 10, 249--256 (1975; Zbl 0304.10029)] showed that \(V(x,q) \sim x\log q\) for almost all \(q\) in the range \({1\over2}Q < q \leq Q\) when \(x/\log^ A x < Q \leq x\). A stronger result conditional on the Generalised Riemann Hypothesis (GRH) was improved, subject to the same hypothesis, by \textit{J. B. Friedlander} and \textit{D. A. Goldston} [Q. J. Math., Oxf. (2) 47, No. 187, 313--336 (1996; Zbl 0859.11054)] to one holding in the range \(x^ {3/4+\varepsilon}<Q \leq x\). They actually obtained a bound for \({\sum_ {Q/2<q\leq Q}}| V(x,q)-U(x,q)| \), in which \(U(x,q)\) is a refined main term: \(U(x,q) = x\log q -CX\), with \(C= \gamma +\log 2\pi +{\sum_ {p\mid q}}(\log p)/(p-1)\). Their bound would be \(\ll x^ 2/\log^ A x\) in the absence of the Riemann Hypothesis.NEWLINENEWLINEThe author considers \(M_ k(x,Q) = {\sum_ {Q/2<q\leq Q}}{| V(x,q)-U(x,q)| }^ k\) for positive integers \(k\). He establishes the inequality NEWLINE\[NEWLINE{M_ k}(x,Q) ={O_ k}( Qx^ k{(F(x/Q)}^ k) +{O_ {k,\varepsilon}}(Qx^ k{(x/Q)}^ kx^ {\varepsilon-1/2})NEWLINE\]NEWLINE when \(x/{\log^ A x} \leq Q \leq x\), with \(F(y) \ll {y^ {-1/2}\exp(-c(\log y)^ {3/5}/(\log\log y)^ {1/5})}\). A stronger result is deduced from the GRH.NEWLINENEWLINEThe method used involves a treatment of the underlying additive problem by a variant of the Hardy-Littlewood method.