Nair-Tenenbaum bounds uniform with respect to the discriminant (Q2883235)

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 NEWLINE\[NEWLINE F(a_1b_1, \dots, a_kb_k) \leq \min\big( A^{\Omega(a_1 \cdots a_k)}, B(a_1 \cdots a_k)^{\varepsilon} \big) F(b_1, \dots, b_k) NEWLINE\]NEWLINE 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 NEWLINE\[NEWLINE S(x, y; F, P_1, \dots P_k) = \sum_{x < n \leq x+y} F(|P_1(n)|, \dots, |P_k(n)|), NEWLINE\]NEWLINE 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 \textit{M. Nair} and \textit{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 \(\Delta\).NEWLINENEWLINEIf \(Q(X) \in \mathbb Z[X]\), 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 NEWLINE\[NEWLINE \begin{aligned} S(x, y; F, P_1, \dots P_k) \ll y & \prod_{p \leq x} \left( 1 - \frac {\rho_P(p)}p \right) \\ & \times K_{\Delta}\sum_{_{\substack{ n_1 \cdots n_k \leq x\\ (n_1 \cdots n_k, \Delta) = 1 }}} F(n_1, \dots, n_k) \frac {\rho_{P_1}(n_1) \cdots \rho_{P_k}(n_k)}{n_1 \cdots n_k}, \end{aligned} NEWLINE\]NEWLINE 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 NEWLINE\[NEWLINE K_{\Delta} = \prod_{p \mid \Delta} \bigg( 1 + \sum_{\nu \leq d} F(p^{\nu}) \left( \frac {\rho_P(p^{\nu})}{p^{\nu}} - \frac {\rho_P(p^{\nu+1})}{p^{\nu+1}} \right) \bigg). NEWLINE\]