1 Introduction

In this paper, we use tools from analytic number theory to estimate the number of triples of a given height satisfying the \(abc\) conjecture. Associated to any non-zero integer \(n\) is its radical

\begin{align*} \operatorname{rad}(n)=\prod _{p\mid n}p. \end{align*}

We say that a triple \((a,b,c)\in \mathbb {N}^3\) with \(\gcd (a,b,c)=1\) is an \(abc\) triple of exponent \(\lambda \) if

\begin{align*} a+b=c,\quad \operatorname{rad}(abc) {\lt} c^{\lambda }. \end{align*}

The well-known \(abc\) conjecture of Masser and Oesterlé asserts that, for any \(\lambda {\lt}1\), there are only finitely many \(abc\) triples of exponent \(\lambda \). The best unconditional result is due to Stewart and Yu [ 12 ] , who have shown that finitely many \(abc\) triples satisfy \(\operatorname{rad}(abc) {\lt} (\log c)^{3-\varepsilon }\). Recently, Pasten [ 11 ] has proved a new subexponential bound, assuming that \(a{\lt}c^{1-\varepsilon }\), via a connection to Shimura curves. In this paper we shall focus on counting the number \(N_\lambda (X)\) of \(abc\) triples of exponent \(\lambda \) in a box \([1,X]^3\), as \(X\to \infty \).

Definition 1.1

✓

For \(\lambda {\gt}0\) define \(N_\lambda (X)\) as the number of triples \((a,b,c)\in \mathbb {N}^3\) with \(a+b=c\), \(\gcd (a,b,c)=1\) and \(\operatorname{rad}(abc) {\lt} c^{\lambda }\).

Given \(\lambda {\gt}0\), an old result of de Bruijn [ 4 ] implies that

Lemma 1.2

For any \(\varepsilon {\gt}0\), we have

\begin{equation} \label{eq:bruijn} \# \left\{ n\leq x : \operatorname{rad}(n)\leq x^\lambda \right\} \ll _\varepsilon x^{\lambda +\varepsilon }. \end{equation}

1.0.1

Proof ▶

It suffices to show that for any integer \(k\geq 2\) we have

\begin{align} \label{eq1} |\{ n\leq X:\, \, \operatorname{rad}(n)=k\} |\ll X^{O(1/\log \log X)}. \end{align}

To prove 1.0.2, write \(k=p_1\cdots p_r\) as distinct primes \(p_1{\lt}\cdots {\lt}p_r\). Then \(\operatorname{rad}(n)=k\) implies that \(n=p_1^{m_1}\cdots p_{r}^{m_r}\) for some integers \(m_1,\ldots , m_r\geq 1\). Therefore, the number of \(n\leq X\) in question is

\begin{align*} |\{ n\leq X:\, \, \operatorname{rad}(n)=k\} | & \leq \sum _{m_1,\ldots , m_r\geq 1}\mathbf{1}_{\sum _{j\leq r}m_j(\log p_j)\leq \log X}\\ & \leq \sum _{m_1,\ldots , m_r\geq 1}\int _{\mathbb {R}^r}\mathbf{1}_{\sum _{j\leq r}t_j(\log p_j)\leq \log X}\mathbf{1}_{t_j\in (m_j-1,m_j]\, \forall j\leq r} \d t_1\cdots \d t_r\\ & =\textnormal{vol}\Big(\big\{ (t_1,\ldots , t_r)\in \mathbb {R}_{\geq 0}^r:\, \, t_1(\log p_1)+\cdots +t_r(\log p_r)\leq \log X\big\} \Big)\\ & =(\log X)^re^{O(r)}r^{-r}\prod _{j=1}^r \frac{1}{\log p_j} \end{align*}

by calculating the volume of a simplex. Moreover, we have

\begin{align*} \prod _{j=1}^r \log p_j\geq \prod _{j=2}^{r+1}\log j=\exp \left(\sum _{j=2}^{r+1}\log \log j\right)\gg (\log r)^{r}e^{-O(r)}. \end{align*}

Hence for some \(C_0{\gt}1\)

\begin{align*} |\{ n\leq X:\, \, \operatorname{rad}(n)=k\} | \ll \left(\frac{C_0\log X}{r\log r}\right)^r\ll \exp \Big(O\Big(\frac{\log X}{\log \log X}\Big)\Big), \end{align*}

since \(r=\omega (k)\leq (1+o(1))(\log X)/(\log \log X)\) by the prime number theorem.

Any triple \((a,b,c)\) counted by \(N_\lambda (X)\) must satisfy \(\operatorname{rad}(abc){\lt} X^{\lambda }\), and so we must have \(\min \{ \operatorname{rad}(a)\operatorname{rad}(b),\operatorname{rad}(b)\operatorname{rad}(c),\operatorname{rad}(c)\operatorname{rad}(a)\} {\lt} X^{2\lambda /3}\), since \(a,b,c\) are pairwise coprime. An application of 1.0.1 now leads to the following “trivial bound”.

Proposition 1.3

Let \(\lambda {\gt}0\). Then \(N_\lambda (X)=O_\varepsilon ( X^{2\lambda /3+\varepsilon })\), for any \(\varepsilon {\gt}0\).

Proof ▶

The primary goal of this paper is to give the first power-saving improvement over this simple bound for values of \(\lambda \) close to \(1\).

Theorem 1.4

Let \(\lambda \in (0, 1.001)\) be fixed. Then \(N_\lambda (X)=O( X^{33/50})\).

Here we note that \(33/50=0.66\). By comparison, the trivial bound in Proposition 1.3 would give \(N_1(X)=O(X^{0.66\bar{6}+\varepsilon })\) and \(N_{1.001}(X)=O(X^{0.6674})\). Moreover, we see that Theorem 1.4 gives a power-saving when \(\lambda \in (0.99,1.001)\). We emphasise that this power-saving represents a proof of concept of the methods; we expect that the exponent can be reduced with substantial computer assistance.

Theorem 1.4 also applies for \(\lambda \) slightly greater than \(1\), which places it in the realm of a question by Mazur [ 10 ] . Given a fixed \(\lambda {\gt}1\), he asked whether or not \( N_\lambda (X)\) has exact order \( X^{\lambda -1}\). In fact, Mazur studies the refined counting function

Definition 1.5

For \(\alpha ,\beta ,\gamma {\gt}0\), define \(S_{\alpha ,\beta ,\gamma }(X)\) as the number of \((a,b,c)\in \mathbb {N}^3\) with \(\gcd (a,b,c)=1\) such that

\[ a,b,c\in [1,X], \quad a+b=c, \quad \operatorname{rad}(a)\leq X^\alpha , \quad \operatorname{rad}(b)\leq X^\beta , \quad \operatorname{rad}(c)\leq X^\gamma . \]

The argument used to prove Proposition 1.3 readily yields

Lemma 1.6

\begin{equation} \label{eq:trivial} S_{\alpha ,\beta ,\gamma }(X)\ll _\varepsilon X^{\min \{ \alpha +\beta , \alpha +\gamma , \beta +\gamma \} +\varepsilon }, \end{equation}

1.0.3

for any \(\varepsilon {\gt}0\).

Proof ▶

Mazur then asks whether \(S_{\alpha ,\beta ,\gamma }(X)\) has order \(X^{\alpha +\beta +\gamma -1}\) if \(\alpha +\beta +\gamma {\gt}1\). Evidence towards this has been provided by Kane [ 9 , Theorems 1 and 2 ] , who proves that

\[ X^{\alpha +\beta +\gamma -1-\varepsilon }\ll _\varepsilon S_{\alpha ,\beta ,\gamma }(X)\ll _\varepsilon X^{\alpha +\beta +\gamma -1+\varepsilon }+X^{1+\varepsilon }, \]

for any \(\varepsilon {\gt}0\), provided that \(\alpha ,\beta ,\gamma \in (0,1]\) are fixed and satisfy \(\alpha +\beta +\gamma {\gt}1\). This result gives strong evidence towards Mazur’s question when \(\alpha +\beta +\gamma \geq 2\), but falls short of the trivial bound 1.0.3 when \(\alpha +\beta +\gamma {\lt}3/2\).

When considering \(abc\) triples of exponent \(\lambda {\lt}1\), we always have \(\alpha +\beta +\gamma \leq \lambda {\lt} 1\), and the methods of Kane give no information in this regime. Indeed, we are not aware of any general estimates when \(\lambda {\lt}1\), beyond Proposition 1.3. Nonetheless, there do exist specific Diophantine equations which are covered by the \(abc\) conjecture and where bounds have been given for the number of solutions. For example, it follows from work of Darmon and Granville [ 5 ] that there are only finitely many coprime integer solutions to the Diophantine equation \(x^p+y^q=z^r\), when \(p,q,r\in \mathbb {N}\) are given and satisfy \(1/p+1/q+1/r{\lt}1\).

Proof outline

We now describe the main ideas behind the proof of Theorem 1.4. In terms of the counting function \(S_{\alpha ,\beta ,\gamma }(X)\), our task is to show that whenever \(\alpha ,\beta ,\gamma \in (0,1]\) satisfy \(\alpha +\beta +\gamma \leq \lambda \), we have \( S_{\alpha ,\beta ,\gamma }(X)\ll X^{2\lambda /3-\eta }, \) for some \(\eta {\gt}0\). A simple factorisation lemma (Proposition 2.6) will reduce the problem of bounding \(S_{\alpha ,\beta ,\gamma }(X)\) to the problem of bounding the number of solutions to various Diophantine equations of the shape

\[ \prod _{j\leq d}x_j^j+\prod _{j\leq d}y_j^j=\prod _{j\leq d}z_j^j , \]

with specific constraints \(x_i\sim X^{a_i}\), \(y_i\sim X^{b_i}, z_i\sim X^{c_i}\) on the size of the variables, for admissible values of \(a_i,b_i,c_i\) (depending on \(\alpha ,\beta ,\gamma \)). We then bound the number of solutions to these Diophantine equations using four different methods. The first of these (Proposition 3.1) uses Fourier analysis and Cauchy-Schwarz to estimate the number of solutions, leading to a bound that works well if two of the exponent vectors \((a_i)_i,(b_i)_i,(c_i)_i\) are somewhat “correlated”. The second method (Proposition 3.2) uses the geometry of numbers and gives good bounds when one of \(a_1,b_1,c_1\) is large. The remaining tools come from the determinant method of Heath-Brown (Proposition 3.14) and uniform upper bounds for the number of solutions to Thue equations (Proposition 3.15). For every choice of the exponents \(a_i,b_i,c_i\) we shall need to take the minimum of these bounds, which leads to a rather intricate combinatorial optimisation problem. This is solved by showing that at least one of the four methods always gives a power-saving over Proposition 1.3 when \(\lambda \) is close to \(1\).

Notation

We shall use \(x\sim X\) to denote \(x\in [X,2X]\) and we put \([d]=\{ 1,\dots ,d\} \). We denote by \(\tau (n)=\sum _{d\mid n}1\) the divisor function.