# Averaging $2^{\omega(n)}$ over a region

Let $R(X)$ be the region defined by

$$\displaystyle R(X) = \{(a,b,c) \in \mathbb{R}^3 \colon |b| \leq a \leq c, \, a,c \geq 1, \, a(4ac-b^2) \leq X\}.$$

I want to know how to estimate the sum

$$\displaystyle \sum_{(a,b,c) \in R(X) \cap \mathbb{Z}^3} 2^{\omega(4ac-b^2)},$$

where $\omega(n)$ denotes the number of prime divisors of $n$. An upper bound suffices.

One can show that

$$\displaystyle \sum_{(a,b,c) \in R(X) \cap \mathbb{Z}^3} 1 = O (X \log X),$$

and it's known that the average of $2^{\omega(n)}$ over $n \leq X$ is of order $\log X$. Therefore, the expected upper bound should be about $O(X (\log X)^2)$.

• It would be nice to mention what is the meaning of $R(X)$ is terms of definite binary quadratic forms. Oct 31, 2017 at 10:09

Your guess is correct. Since $2^{\omega(n)} = \sum_{d | n} \mu^2(d)$ one has $$S = \sum_{(a,b,c) \in R(X)} 2^{\omega(4ac-b^2)} = \sum_{d \leq X} \mu^2(d) |R_d(X)|$$ where $R_d(X)$ is the set of integer triples $(a,b,c) \in R(X)$ such that $4ac \equiv b^2 \pmod d$.

Let $d$ be a squarefree integer. Let $a,b$ be such that $|b| \leq a$ and let $s = \mathrm{gcd}(4a,d)$. Then for $(a,b,c)$ to be in $R_d(X)$ one must have $s | b^2$, or equivalently $s |b$ since $s$ is squarefree, and also $c \equiv (4a/s)^{-1} b^2 \ \mathrm{mod} \ d/s$. The number of integers $c$ like that, with $a \leq c \leq b^2/(4a) + X/(4a^2)$ is

$$\leq \frac{s}{d}( \frac{b^2}{4a} + \frac{X}{4a^2} - a) + 1 \leq \frac{s}{d}( \frac{X}{4a^2} - \frac{3 a}{4}) + 1\leq\frac{sX}{4 a^2d} + \mathrm{1}_{as \leq 4d/3}.$$

Moreover the number of integers $b$ such that $|b| \leq a$ and $s | b$ is $$\leq \frac{2a}{s} + 2.$$ Since the constraint $X \geq a(4ac-b^2)$ yields $ad \leq X$, we have $$|R_d(X)| \ll \sum_{ad \leq X} (\frac{sX}{a^2d} + \mathrm{1}_{as \leq 4d/3})(\frac{a}{s} + 1) \\ \ll \sum_{ad \leq X} ( \frac{X}{ad} + \frac{sX}{a^2d} + \frac{a}{s} \mathrm{1}_{as \leq 4d/3} + 1 ) \ll \frac{X \log X}{d} + \frac{X}{d} + \sum_{ ad \leq X} ( \frac{sX}{a^2d} + \mathrm{1}_{as \leq d} \frac{a}{s} ).$$ Summing over $d$ we get $$S \ll X \log^2(X) + \sum_{s} \sum_{ad \leq X \\ s|4a , s|d} ( \frac{sX}{a^2d} + \mathrm{1}_{as \leq 4d/3} \frac{a}{s} ).$$ For each $s$, the inner sum is $$\ll \sum_{ad \leq X/s^2 } ( \frac{X}{s^2 a^2d} + \mathrm{1}_{as \leq 16d/3} a )$$ The first sum is $\ll X s^{-2}$, while in the second we have the constraint $a^2 \ll X s^{-3}$ which yields $$\ll \sum_{a \ll X^{1/2} s^{-3/2}} a \ll X s^{-3} + X^{1/2} s^{-3/2}.$$ So the global sum over $s$ is $\ll X$ and we are done.

Remark: one should be able to extract from this proof an asymptotic formula of the form $c X \log^2(X) + O(X \log X)$.

• There seems o be a mistake in the inequality $\frac{s}{d}( \frac{X}{4a^2} - \frac{3 a}{4}) + 1\leq\frac{sX}{4 a^2d} + \mathrm{1}_{as \leq d}.$. Namely, if $d<as<4d/3$ then the inequality fails. Oct 31, 2017 at 13:01
• Oh, absolutely. It should be $as \leq 4 d/3$ instead. It does not really affect the proof though.
– js21
Oct 31, 2017 at 13:16
• Well it does not affect the proof but was apparently worth a downvote :)
– js21
Oct 31, 2017 at 13:20
• cool, i corrected all typos. There were a few more other than $as\leq 4d/3$. Oct 31, 2017 at 13:51
• not sure if my edits were saved. One more typo is:The first sum is ≪Xs−2 X s 2 , while --->The first sum is ≪Xs−2 \log X X s 2 , while Oct 31, 2017 at 13:53