1. Remark on the Farey fraction spin chain.
- Author
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Technau, Marc
- Subjects
- *
NUMBER theory , *FRACTIONS , *MATRIX multiplications , *L-functions , *ENERGY policy , *MATHEMATICS - 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 \Phi (N) of 2{\times }2 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 [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 \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 \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. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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