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Distribution of arithmetical functions in the mean with respect to progressions (theorems of Vinogradov-Bombieri type). (Russian) Zbl 0566.10037

The authors prove some generalizations of the Bombieri-Vinogradov theorem. Let \[ \sum (x,f,k,\ell)=\sum_{n\leq x;n\equiv \ell (mod k)}f(n)-(1/\phi (k))\sum_{n\leq x;(n,k)=1}f(n), \]
\[ \delta (x,f,k)=\max_{(\ell,k)=1}\max_{y\leq x}| \sum (y,f,k,\ell)|,\quad \Delta (Q,f,E)=\sum_{k\leq Q,k\in E}\delta (x,f,k) \] F(k;E) be the characteristic function of the set E, \[ \Delta_ 1(Q,f,E)=\sum_{k\leq Q}F(k,E)\max_{(\ell,k)=1}\max_{y\leq x}| \]
\[ \sum_{p\leq y;p\equiv \ell (mod k)}f(p) \log p-(1/\phi (k))\sum_{p\leq y;p\nmid k}f(p) \log p|, \] f*g(n)\(=\sum_{dm=n}f(d) g(m)\), \(\hat g(n)\) and \(\Lambda_ g(n)\) be functions such that \(g*\hat g=e\) and \(g(n)\log n=(g*\Lambda_ g)(n)\) respectively; \(g_ u(n)=\sum_{dm=u;d>u}\hat g(d) g(m)\); \(M(f,x)=\sum_{n\leq x}f(n)\); \(L=\log x\), \(Q_ i=Q_ i(x)\) be monotone - increasing to \(\infty\) functions. Let \(\mu_{\alpha}(D)\) be the class of multiplicative functions such that \(M(| f|^ 4,x)\ll x L^{4\alpha}\), \(\alpha\geq 0\) and for all primitive characters mod d, \(d\in D\), \(d<L^ c\ell\) we have \(\sum_{z<p\leq y}\chi^*_ d(p) f(p) \log p\ll y L^{-B}\ell\), where \(z=\exp (L^{\theta})\), \(\theta =1- (\log \log L/\log L)\), \(y\leq x\), \(C_ 1,B_ 1\) are some constants, \(D\subset z^+\). The following main results (among many other results) are obtained:
Theorem 1. Let \(g(1)\neq 0\) and let (1) \(M(| \hat g|^ 4,x)\ll x L^{4\alpha},M(| g_ n|^ 4,x)\ll x L^{4\alpha} \log^{4\beta}u\) for \(u\geq 2\), \(\alpha,\beta,\gamma >0\). We assume that there exists \(Q_ 1(x)\) such that (2) \(\Delta(Q_ 1,g,x)\ll x L^{-A}\) and (3) \(\sum_{n\leq x;(n,m)=1}\chi_ q(u) \hat g(u)\ll x \exp (- L^{\sigma}) \log m\), \(\sigma >0\) for all primitive characters \(\chi\) mod q and all m, where \(q\leq L^ c \ell\), \(c_ 1>0\). Then, if \(C\leq \min \{A-\delta (\alpha +1)-1\), \(A_ 2-2-\alpha -\gamma\), \(C_ 1-\alpha -\gamma -2\}-\epsilon\), \(\epsilon >0\), \(0<\delta <1\), Q(x)\(\leq \min \{Q_ 1(x L^{-2A_ 2}\exp (L^{\delta})\), \(\sqrt{x} L^{-c-\alpha - \gamma -1} \log^{-\beta -1}L\}\), we have \(\Delta(Q,\hat g,N)\ll x L^{-c}.\)
Theorem 2. Let \(g(1)\neq 0\) and let \(g(n)\) be a function satisfying (1), (2) and (3) with \(\Lambda_ g(u)\) instead of \(\hat g(\)u). Let \(M(| \Lambda_ g|^ 4,x)\ll x L^{4\alpha}\). Then \(\Delta (Q,\Lambda_ q,N)\ll x L^{-c}\), where Q and c are the same as in Theorem 1.
Theorem 3. (1) Let \(f(u)\) be a fully multiplicative function, \(f\in M_{\alpha}(D)\), \(\Delta (Q_ 1,f,E)\ll x L^{-A}\). Then \(\Delta_ 1(Q,f,E)\ll x L^{-A+1} \log^{\alpha +1}L\), where Q(x)\(\leq \min (Q_ 1(x L^{-2A-4\alpha -7/2} \log^{-2\alpha -26}L)\), \(\sqrt{x} L^{-A- 2\alpha -3/4} \log^{-\alpha -13}L)\) and E is a subset of natural numbers having all its divisors \(\in D\). (2) If \(Q_ 1(x)\geq x^{1/2+\epsilon}\) and \(f(u)=O(1)\), then \(\Delta_ 1(Q,f,E)\ll x L^{- A+2}\) for \(Q=\sqrt{x} L^{-A+1} \log^{-3/2}L\). If \(f(u)=1\), then for \(Q=\sqrt{x} L^{-A-1} \log^{-3/2}L\) we have \[ \sum_{k\leq Q}\max_{(\ell,k)=1}\max_{y\leq x}| \psi (y,k,\ell)-\frac{y}{\phi (k)}| \ll x L^{-A}, \] Theorem 4. Let \(f\in M_{\alpha}(D)\) and \(\Delta_ 1(Q,f,E)\ll x L^{-3B}\). Then \[ \Delta (Q_ 1,f,E)<\ll x L^{-B+5/6+4/3\alpha} Log^{\alpha +2}L \] with \(Q_ 1=\min (Q(x),\sqrt{x} L^{-3B-3/2-2\alpha} Log^{-5/4}L)\). Some interesting corollaries are also proved.
Reviewer: G.A.Kolesnik

MSC:

11N37 Asymptotic results on arithmetic functions
11N13 Primes in congruence classes
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