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Short sums of certain arithmetic functions. (English) Zbl 0917.11048
In [J. Reine Angew. Math. 313, 161-170 (1980; Zbl 0412.10030)] P. Shiu obtained a uniform upper bound for the sum over an arithmetic progression in a short interval of a multiplicative function in a certain class, and this result was later generalized by M. Nair. The object of the present paper is to establish a substantial generalization of these results, which has wide applications and leads to new estimates.
For a fixed positive integer $$k$$, let $$M_k(A,B,\varepsilon)$$ denote the class of non-negative arithmetic functions $$F(n_1,\dots,n_k)$$ such that $$F(m_1n_1,\dots, m_kn_k)\leq\min (A^{\Omega(m)}, Bm^\varepsilon)F(n_1,\dots,n_k)$$ for all $$k$$-tuples $$(m_1, \dots, m_k)$$, $$(n_1, \dots, n_k)$$ with $$(m_j,n_j)=1$$ for $$1\leq j\leq k$$, where $$m=m_1\dots m_k$$. This covers a wide class of functions, which need not even be sub-multiplicative; for example, Hooley’s $$\Delta$$-function belongs to $$M_1(2,B, \varepsilon)$$ for any $$\varepsilon>0$$ and some $$B=B(\varepsilon)$$. Suppose that $$Q_j\in\mathbb{Z}[X]$$ for $$1\leq j\leq k$$, where $$Q=Q_1\dots Q_k$$ has no fixed prime divisor. The authors establish an upper bound for $$\sum_{x<n\leq x+y}F(| Q_1(n) |,\dots,| Q_k(n)|)$$ that is uniform for $$x^\alpha\leq y\leq x$$ and $$x$$ sufficiently large, where $$F\in M_k(A,B,\varepsilon)$$ for any $$A\geq 1$$, $$B\geq 1$$ and sufficiently small $$\varepsilon>0$$, and where $$\alpha=\alpha(\varepsilon)$$ is of a specified form. In theorem 3, they consider the analogous problem when $$Q(0) \neq 0$$ and $$n$$ is restricted to prime values. When $$Q\in \mathbb{Z}[X]$$ is irreducible, theorem 2 provides a lower bound for the largest prime factor of $$\prod_{x<n\leq x+y}Q(n)$$, where $$y=x^\beta$$ for $$\beta\leq 1$$ of a given form.
A number of useful and illuminative corollaries are given, and we pick out two striking special cases featuring well known functions. Write $$L(z)=\exp (\sqrt {\log z\log_2z})$$, and let $$0<\varepsilon<1$$. Corollaries 5 and 9 state respectively that uniformly for $$D\geq 1$$, $$x^\varepsilon\leq y\leq x$$ $\sum_{D <d\leq 2D}\left( \left[{x+y \over d}\right] -\left[{x\over d} \right]\right) \ll y \bigl(L(\log x)\bigr)^{\sqrt 2+o(1)}\text{ as }x\to\infty;\tag{i}$
$\max_{D \in \mathbb{R}}\sum_{D<q\leq 2D}\bigl(\pi(x+y;q,a)-\pi(x;q,a)\bigr)\ll{y\over \log x} \bigl(L(\log x)\bigr)^{\sqrt 2+o(1)}. \tag{ii.}$

##### MSC:
 11N37 Asymptotic results on arithmetic functions 11N56 Rate of growth of arithmetic functions 11N64 Other results on the distribution of values or the characterization of arithmetic functions
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##### References:
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