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Moments for primes in arithmetic progressions. I, II. (English) Zbl 1053.11078

The Barban-Davenport-Halberstam theorem concerns the mean-values \[ \sum_{q\leq Q}\;\sum_{a\leq q,(a,q)=1} \Biggl\{\psi(x; q,a)- {x\over\phi(q)}\Biggr\}^2. \] The purpose of these papers is to replace \(x/\phi(q)\) by the approximation \[ \rho(x;q,a)= \sum_{n\leq x,n\equiv a\pmod q} F_R(n), \] with \[ F_R(n)= \sum_{r\leq R} {\mu(r)\over \phi(r)} \sum_{b\leq r,(b,r)= 1} e(bn/r). \] It is well known that \(F)R(n)\) mimics \(\Lambda(n)\) fairly well, see D. A. Goldston [Expo. Math. 13, No. 4, 366–376 (1995; Zbl 0854.11044)], and the conclusion of the first of these two papers is that
\[ \sum_{q\leq Q}\;\sum_{a\leq q,(a,q)= 1} \{\psi(x;q,a)- \rho(x; q,a)\}^2= Qx\log\bigl(\tfrac x2\bigr)- cQx+ O(QxR^{-1/2})+ O(x^2 R^{-1}\log^2 x) \]
for a suitable constant \(c\), uniformly for \(Q\leq x\) and \(R\leq(\log x)^4\). Not only is the main term smaller than in the Barban-Davenport-Halberstam theorem, but the error term remains non-trivial even as \(Q\) approaches \(x\). A number of other results describing the ways in which \(\rho\) approximates \(\psi\) are also given.
In the second paper the third moment \[ \sum_{q\leq Q}q \sum_{a\leq q,(a,q)= 1} \{\psi(x; g,a)- \rho(x;q,a)\}^3 \] is similarly investigated, and the asymptotic formula
\[ \tfrac12 Q^2x\log^2x- \tfrac 32 Q^2 x(\log x)(\log R+ c)+ O(x^3 R^{-1}\log^5 x)+ O(Q^2 x\log^3 R)+ O(Q^2 xR^{-1/2}\log x) \]
is obtained. This should be compared with the result of C. Hooley [J. Reine Angew. Math. 499, 1–46 (1998; Zbl 0907.11028)] for the third moment in the standard version of the Barban-Davenport-Halberstam theorem. The present paper again gives sharper estimates for \(Q\) comparable to \(x\).

MSC:

11N13 Primes in congruence classes
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