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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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##### References:
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