Remark on the Farey fraction spin chain

In 1999, Kleban and Özlük introduced a `Farey fraction spin chain’ and made a conjecture regarding its asymptotic number of states with given energy, the latter being given (up to some normalisation) by the number $\mathrm{\Phi }(N)$ of $2\times 2$ matrices arising as products of $\left(\text{\hspace{0.17em}⁣}\begin{array}{cc}1& 0\\ 1& 1\end{array}\text{\hspace{0.17em}⁣}\right)$ and $\left(\text{\hspace{0.17em}⁣}\begin{array}{cc}1& 1\\ 0& 1\end{array}\text{\hspace{0.17em}⁣}\right)$ whose trace equals $N$. Although their conjecture was disproved by Peter (2001), quite precise results are known on average by works of Kallies–Özlük–Peter–Snyder (2001), Boca (2007) and Ustinov (2013). We show that the problem of estimating $\mathrm{\Phi }(N)$ can be reduced to a problem on divisors of quadratic polynomials which was already solved by Hooley (1958) in a special case and, quite recently, in full generality by Bykovski{ĭ} and Ustinov (2019). This produces an unconditional estimate for $\mathrm{\Phi }(N)$, which hitherto was only (implicitly) known, conditionally on the availability on wide zero-free regions for certain Dirichlet $L$-functions, by the work of Kallies–Özlük–Peter–Snyder.