Remark on the Farey fraction spin chain

Abstract

Kleban and Özlük [Comm. Math. Phys. 203 (1999), pp. 635–647] 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 Φ ( N ) \Phi (N) of 2 × 2 2{\times }2 matrices arising as products of ( 1 a m p ; 0 1 a m p ; 1 ) \bigl (\begin {smallmatrix} 1 & 0 \\ 1 & 1 \end {smallmatrix}\bigr ) and ( 1 a m p ; 1 0 a m p ; 1 ) \bigl (\begin {smallmatrix} 1 & 1 \\ 0 & 1 \end {smallmatrix}\bigr ) whose trace equals N N . Although their conjecture was disproved by Peter [J. Number Theory 90 (2001), pp. 265–280], quite precise results are known on average by works of Kallies–Özlük–Peter–Snyder [Comm. Math. Phys. 203 (1999), pp. 635–647], Boca [J. Reine Angew. Math. 606 (2007), pp. 149–165] and Ustinov [Mat. Sb. 204 (2013), pp. 143–160]. We show that the problem of estimating Φ ( N ) \Phi (N) can be reduced to a problem on divisors of quadratic polynomials which was already solved by Hooley [Math. Z. 69 (1958), pp. 211–227] in a special case and, quite recently, in full generality by Bykovskiĭ and Ustinov [Dokl. Math. 99 (2019), pp. 195–200]. This produces an unconditional estimate for Φ ( N ) \Phi (N) , which hitherto was only (implicitly) known, conditionally on the availability on wide zero-free regions for certain Dirichlet L L -functions, by the work of Kallies–Özlük–Peter–Snyder.