Number of polynomial values dividing an integer (Q2760477)

Let \(P\) be a quadratic polynomial with integer coefficients and discriminant \(\Delta\), and let \(\tau_P(n)= \text{card} \{P(k)> 0: P(k)\mid n,k\in \mathbb{Z}\}\). Thus when \(P(X)= x(X+1)\), \(\tau_P(n)\) counts the number of consecutive divisors of \(n\), a quantity that has been studied by various authors [see the references cited in the paper, for example Analytic Number Theory, Urbana, 1989, Prog. Math. 85, 77-90 (1990; Zbl 0718.11041) by \textit{A. Balog, P. Erdős} and \textit{G. Tenenbaum}]. NEWLINENEWLINENEWLINELet \(D(n)= 2^{(\log n)/ (\log\log n)}\) for \(n\geq 3\); then it is well known that \(\max_{m\leq n} \tau(m)= (D(n))^{1+o(1)}\). The author proves in Theorem 1 that NEWLINE\[NEWLINE\begin{alignedat}{2} \max_{m\leq n} \tau_P(m) &= (D(n))^{\frac 12+ o(1)} &&\quad\text{when }\Delta=0,\\ \max_{m\leq n} \tau_P(m) &\leq (D(n))^{c+o(1)} &&\quad\text{when }\Delta\neq 0. \end{alignedat}NEWLINE\]NEWLINE The value of \(c\), which is given explicitly, depends on whether \(\Delta\) is a nonzero square, or is not a square, in \(\mathbb{Z}\); in either case \(c> \frac 12\). NEWLINENEWLINENEWLINEThe author also establishes, and uses in the proof of Theorem 1, upper bounds for the quantities NEWLINE\[NEWLINE\begin{aligned} \text{card} &\{d\mid n: (d,s)=1,\;d(d+s)\mid n\},\\ \text{card} &\{d\mid n: d\leq n^\alpha\} \quad\text{when }\alpha\geq 0. \end{aligned}NEWLINE\]NEWLINE When \(\Delta\) is not a square in \(\mathbb{Z}\), the problem of Theorem 1 is transformed into one involving ideals of the ring of integers of the field \(\mathbb{Q}(\sqrt{\Delta})\), and then similar techniques to those used for the case \(\Delta\) a nonzero square are applied.