Remark on the Farey fraction spin chain

In 1999, Kleban and \"Ozl\"uk 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 \(\Phi(N)\) of \(2\times2\) matrices arising as products of \(\bigl(\!\begin{smallmatrix} 1 & 0 \\ 1 & 1 \end{smallmatrix}\!\bigr)\) and \(\bigl(\!\begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix}\!\bigr)\) whose trace equals \(N\). Although their conjecture was disproved by Peter (2001), quite precise results are known on average by works of Kallies--\"Ozl\"uk--Peter--Snyder (2001), Boca (2007) and Ustinov (2013). We show that the problem of estimating \(\Phi(N)\) 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{\uı} and Ustinov (2019). This produces an unconditional estimate for \(\Phi(N)\), 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--\"Ozl\"uk--Peter--Snyder.