## 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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