Sums of arithmetic functions over values of binary forms.

*(English)*Zbl 1159.11035There are a number of papers in the literature dealing with upper bounds for sums of the form
\[
\sum_{n\leq X} h(F(n))
\]
and their generalizations, where \(h(*)\) is a nonnegative multiplicative, (or even sub-multiplicative) function, and \(F(*)\) is a polynomial. The most advanced of these is the work by M. Nair and G. Tenenbaum [Acta Math. 180, No. 1, 119–144 (1998; Zbl 0917.11048)]. The purpose of the present paper is to establish a result in which \(F\) is replaced by a binary form. The authors use the obvious route, in which one applies the earlier results to the polynomial \(F_y(X):= F(X, y)\), and sums for \(y\leq Y\). In doing so they encounter problems of uniformity in the earlier results, which are successfully overcome.

Given any nonnegative constant \(A\), and any nonnegative function \(B(*)\), one defines the class \(M(A, B)\) of nonnegative sub-multiplicative functions \(h(*)\), by the conditions that \(h(p^e)\leq A^e\) for all \(e\in\mathbb{N}\), and \(h(n)\leq B(\varepsilon) n^\varepsilon\) for all \(n\in\mathbb{N}\) and all \(\varepsilon> 0\). We set \(h(0)= 0\) conventionally. Let \(F(X, Y)\) be a non-zero primitive integral binary form of degree \(d\), without any repeated factor, and with zeros of order \(e\) and \(f\) (in \(\{0,1\}\)) respectively at \((1, 0)\) and \((0, 1)\). Write \(\Delta\) for the discriminant of \(F\), and let \(\rho(p)\) be the number of solutions of \(F(m, n)\equiv\pmod p\), with neither variable divisible by \(p\), up to projective equivalence. Then for any \(\delta\in(0, 1)\) there are positive constants \(c= c(A, B)\) and \(C= C(A,B, d,\delta)\) such that \[ \sum_{m\leq X,n\leq Y} h(|F(m,n)|)\ll_{A,B,\delta}\Biggl({\sigma(\Delta)\over\Delta}\Biggr)^c XYE, \] where \[ E=\prod_{p\leq \min\{X,Y\}} \Biggl(1+{(d+ e)(h(p)- 1)\over p}\Biggr) \prod_{d< p\leq\min\{X, Y\}} \Biggl(1+ {\rho(p)(h(p)- 1)\over p}\Biggr). \] This holds uniformly for \[ \min\{X, Y\}\geq C\max\{X, Y\}^{d\delta}\| F\|^\delta. \] This type of result is useful in certain problems involving counting rational points on algebraic varieties, as in the reviewer’s work [Linear relations amongst sums of two squares. Number theory and algebraic geometry, Cambridge: Cambridge University Press. London Mathematical Society Lecture Note Series 303, 133–176 (2003; Zbl 1161.11387)]. Indeed the authors point out a minor error in the reviewer’s paper, which may be corrected by using the results of the present work.

As mentioned above the proof depends on a result for polynomials. Let \(h\in M(A, B)\) and let \(f\) be an integer polynomial of degree \(d\) and with no fixed prime factor. Then for any \(\delta\in(0, 1)\) there is a constant \(C= C(A,B,d,\delta)\) such that \[ \sum_{m\leq X} h(|f(m)|\ll_{A,B,d,\delta} X\prod_{p\leq X}\Biggl(1- {\rho(p)\over p}\Biggr) \sum_{m\leq X} {h(m)\rho(m)\over m} \] uniformly for \(X\geq c\| f\|^\delta\), where \(\rho(m)\) is now the number of zeros off modulo \(m\).

The significance of this second result is that, unlike its predecessors, it is independent of any factor depending on the discriminant of \(f\). The proof is not given, and indeed the suggestion that it is a special case of a result of M. Nair and G. Tenenbaum [loc. cit.] is misleading, since the latter contains an implicit dependence on a discriminant. However as the authors indicate, one can use the method of M. Nair [Acta Arith. 62, No. 3, 257–269 (1992; Zbl 0768.11038)], with a little extra work, to establish the theorem as quoted.

Given any nonnegative constant \(A\), and any nonnegative function \(B(*)\), one defines the class \(M(A, B)\) of nonnegative sub-multiplicative functions \(h(*)\), by the conditions that \(h(p^e)\leq A^e\) for all \(e\in\mathbb{N}\), and \(h(n)\leq B(\varepsilon) n^\varepsilon\) for all \(n\in\mathbb{N}\) and all \(\varepsilon> 0\). We set \(h(0)= 0\) conventionally. Let \(F(X, Y)\) be a non-zero primitive integral binary form of degree \(d\), without any repeated factor, and with zeros of order \(e\) and \(f\) (in \(\{0,1\}\)) respectively at \((1, 0)\) and \((0, 1)\). Write \(\Delta\) for the discriminant of \(F\), and let \(\rho(p)\) be the number of solutions of \(F(m, n)\equiv\pmod p\), with neither variable divisible by \(p\), up to projective equivalence. Then for any \(\delta\in(0, 1)\) there are positive constants \(c= c(A, B)\) and \(C= C(A,B, d,\delta)\) such that \[ \sum_{m\leq X,n\leq Y} h(|F(m,n)|)\ll_{A,B,\delta}\Biggl({\sigma(\Delta)\over\Delta}\Biggr)^c XYE, \] where \[ E=\prod_{p\leq \min\{X,Y\}} \Biggl(1+{(d+ e)(h(p)- 1)\over p}\Biggr) \prod_{d< p\leq\min\{X, Y\}} \Biggl(1+ {\rho(p)(h(p)- 1)\over p}\Biggr). \] This holds uniformly for \[ \min\{X, Y\}\geq C\max\{X, Y\}^{d\delta}\| F\|^\delta. \] This type of result is useful in certain problems involving counting rational points on algebraic varieties, as in the reviewer’s work [Linear relations amongst sums of two squares. Number theory and algebraic geometry, Cambridge: Cambridge University Press. London Mathematical Society Lecture Note Series 303, 133–176 (2003; Zbl 1161.11387)]. Indeed the authors point out a minor error in the reviewer’s paper, which may be corrected by using the results of the present work.

As mentioned above the proof depends on a result for polynomials. Let \(h\in M(A, B)\) and let \(f\) be an integer polynomial of degree \(d\) and with no fixed prime factor. Then for any \(\delta\in(0, 1)\) there is a constant \(C= C(A,B,d,\delta)\) such that \[ \sum_{m\leq X} h(|f(m)|\ll_{A,B,d,\delta} X\prod_{p\leq X}\Biggl(1- {\rho(p)\over p}\Biggr) \sum_{m\leq X} {h(m)\rho(m)\over m} \] uniformly for \(X\geq c\| f\|^\delta\), where \(\rho(m)\) is now the number of zeros off modulo \(m\).

The significance of this second result is that, unlike its predecessors, it is independent of any factor depending on the discriminant of \(f\). The proof is not given, and indeed the suggestion that it is a special case of a result of M. Nair and G. Tenenbaum [loc. cit.] is misleading, since the latter contains an implicit dependence on a discriminant. However as the authors indicate, one can use the method of M. Nair [Acta Arith. 62, No. 3, 257–269 (1992; Zbl 0768.11038)], with a little extra work, to establish the theorem as quoted.

Reviewer: Roger Heath-Brown (Oxford)