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Sums of arithmetic functions over values of binary forms. (English) Zbl 1159.11035
There 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.

##### MSC:
 11N37 Asymptotic results on arithmetic functions 11N32 Primes represented by polynomials; other multiplicative structures of polynomial values
##### Keywords:
binary form; multiplicative function; average; upper bound; uniform
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