Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Continued Fractions and Hardy Sums   The classical Dedekind sums $s(d, c)$ can be represented as sums over the
partial quotients of the continued fraction expansion of the rational
$\frac{d}{c}$. Hardy sums, the analog integer-valued sums arising in the
transformation of the logarithms of $\theta$-functions under a subgroup of the
modular group, have been shown to satisfy many properties which mirror the
properties of the classical Dedekind sums. The representation as sums of
partial quotients has, however, been missing so far. We define non-classical
continued fractions and prove that Hardy sums can be expressed as a sums of
partial quotients of these continued fractions. As an application, we give a
new proof that the graph of the Hardy sums is dense in $\mathbf{R} \times
\mathbf{Z}$.

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The categories of this paper are: plasm-ph
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The categories of this paper are: math-ph
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The categories of this paper are: q-alg
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The categories of this paper are: astro-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: eess
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The categories of this paper are: nlin
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The categories of this paper are: hep-lat
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The categories of this paper are: q-bio
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The categories of this paper are: cond-mat
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The categories of this paper are: supr-con
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The categories of this paper are: adap-org
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The categories of this paper are: gr-qc
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The categories of this paper are: stat
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The categories of this paper are: comp-gas
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The categories of this paper are: quant-ph