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Nair-Tenenbaum bounds uniform with respect to the discriminant. (English) Zbl 1255.11048
Math. Proc. Camb. Philos. Soc. 152, No. 3, 405-424 (2012); erratum ibid. 157, No. 2, 375-377 (2014).
For fixed constants \(k \geq 1\), \(A \geq 1\), \(B \geq 1\), and \(\varepsilon > 0\), let \(\mathcal M_k(A, B, \varepsilon)\) denote the class of arithmetic functions \(F: \mathbb N^k \to [0, \infty)\) such that \[ F(a_1b_1, \dots, a_kb_k) \leq \min\big( A^{\Omega(a_1 \cdots a_k)}, B(a_1 \cdots a_k)^{\varepsilon} \big) F(b_1, \dots, b_k) \] whenever \(\gcd(a_1 \cdots a_k, b_1 \cdots b_k) = 1\). (Here, \(\Omega(n)\) denotes the number of prime divisors of the positive integer \(n\), counted with multiplicities.) Let \(P_1(X), \dots, P_k(X) \in \mathbb Z[X]\) be irreducible and pairwise coprime, and let \(P(X) = P_1(X) \cdots P_k(X)\) have degree \(d\) and discriminant \(\Delta\). In this paper, the author establishes upper bounds for sums of the form \[ S(x, y; F, P_1, \dots P_k) = \sum_{x < n \leq x+y} F(|P_1(n)|, \dots, |P_k(n)|), \] where \(F \in \mathcal M_k(A, B, \varepsilon)\), \(x\) is large, and \(x^{\alpha} < y \leq x\) for some fixed \(\alpha \in (0,1)\). His results build upon earlier work by M. Nair and G. Tenenbaum [Acta Math. 180, No. 1, 119–144 (1998; Zbl 0917.11048)], but unlike the bounds obtained by those authors, the bounds in the paper under review are explicit in their dependence on the discriminant \(\Delta\).
If \(Q(X) \in \mathbb Z[X]\), let \(\rho_Q(n)\) denote the number of roots of \(Q(X)\) modulo \(n\). Assume that \(\rho_P(p) < p\) for all primes \(p\) (i.e., the values of \(P\) on the integers have no fixed prime divisor) and that \(x\) and \(y\) are as above. Assume also that \(F \in \mathcal M_k(A, B, \varepsilon)\) is multiplicative and that \(\varepsilon\) is sufficiently small (this is quantified in paper). Then a special case of the main result of the paper is the estimate \[ \begin{aligned} S(x, y; F, P_1, \dots P_k) \ll y & \prod_{p \leq x} \left( 1 - \frac {\rho_P(p)}p \right) \\ & \times K_{\Delta}\sum_{_{\substack{ n_1 \cdots n_k \leq x\\ (n_1 \cdots n_k, \Delta) = 1 }}} F(n_1, \dots, n_k) \frac {\rho_{P_1}(n_1) \cdots \rho_{P_k}(n_k)}{n_1 \cdots n_k}, \end{aligned} \] where \(K_{\Delta}\) is a product over the primes \(p \mid \Delta\) and the implied constant depends only on \(d, \alpha, A, B\). For example, when \(k = 1\), the product \(K_{\Delta}\) takes the form \[ K_{\Delta} = \prod_{p \mid \Delta} \bigg( 1 + \sum_{\nu \leq d} F(p^{\nu}) \left( \frac {\rho_P(p^{\nu})}{p^{\nu}} - \frac {\rho_P(p^{\nu+1})}{p^{\nu+1}} \right) \bigg). \]

MSC:
11N37 Asymptotic results on arithmetic functions
11N56 Rate of growth of arithmetic functions
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