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(\!\b...
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Zusammenfassung: | 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{\u{\i}}
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. |
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DOI: | 10.48550/arxiv.2304.08143 |