## 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

### Citations:

Zbl 0854.11044; Zbl 0907.11028
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