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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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