Remark on the Farey fraction spin chain

Abstract

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 $\hat{I}¦(N)$ of $2\times 2$ matrices arising as products of $\left(\begin{array}{cc}1& 0\\ 1& 1\end{array}\right)$ and $\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$ 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 $\hat{I}¦(N)$ 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 $\hat{I}¦(N)$, 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.

7 pages