4
$\begingroup$

The equation $x_1x_2+y_1y_2=n$ is well-studied (Ingham, Heath-Brown, Deshouillers & Iwaniec, Ismoilov) because it arises in an additive divisor problem. The number of solutions in positive integers is $n(c_2\log^2n+c_1\log n+c_0)+O(n^{1-\delta})$ where $c_i$'s are some explicit arithmetic functions of $n$.

The same tools allow to prove similar formula for the number of solutions of the equation $ax_1x_2+by_1y_2=n$. In this case $c_i$'s will depend on $n,a$ and $b$.

Is it possible to find this result in a literature?

Improve this question
$\endgroup$
2
  • $\begingroup$ Do you assume $a,b$ to be positive (integers)? $\endgroup$
    – Dirk
    Commented Jul 27, 2018 at 9:08
  • $\begingroup$ Yes, positive integer constants. $\endgroup$
    – Alexey Ustinov
    Commented Jul 27, 2018 at 9:43

1 Answer 1

Reset to default
7
$\begingroup$

For a smoothened version of your sum, an asymptotic formula can be found in Duke-Friedlander-Iwaniec: A quadratic divisor problem (Inventiones, 1994), see (1)-(5) there. It is straightforward to "unsmooth" this formula, much like it is done in (6)-(7) of the paper.

Improve this answer
$\endgroup$
4
  • $\begingroup$ Are there any results where the sum $\sum_{am+bn=h}\tau(m)\tau(n)f(am,bn)$ is replased by more general one $\sum_{ax_1x_2+by_1y_2=h}f(x_1,x_2,y_1,y_2)$? $\endgroup$
    – Alexey Ustinov
    Commented Jul 28, 2018 at 7:17
  • $\begingroup$ @AlexeyUstinov: I don't know about this being worked out. However, it should be doable with the Kloosterman refinement of the circle method, see e.g. Heath-Brown's Crelle paper for a flexible setup. $\endgroup$
    – GH from MO
    Commented Jul 28, 2018 at 9:46
  • $\begingroup$ Probably you meant this one "A new form of the circle method, and its application to quadratic forms." degruyter.com/view/j/crll.1996.issue-481/crll.1996.481.149/… $\endgroup$
    – Alexey Ustinov
    Commented Jul 29, 2018 at 5:03
  • $\begingroup$ @AlexeyUstinov: Yes. $\endgroup$
    – GH from MO
    Commented Jul 29, 2018 at 9:48

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .