Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Moments of averages of generalized Ramanujan sums   Let $\beta$ be a positive integer. A generalization of the Ramanujan sum due
to Cohen is given by \begin{align} c_{q,\beta }(n) :=
\sum\limits_{{{(h,{q^\beta })}_\beta } = 1} {{e^{2\pi inh/{q^\beta }}}},
\nonumber \end{align} where $h$ ranges over the the non-negative integers less
than $q^{\beta}$ such that $h$ and $q^{\beta}$ have no common $\beta$-th power
divisors other than $1$. The distribution of the average value of the Ramanujan
sum is a subject of extensive research. In this paper, we study the
distribution of the average value of $c_{q,\beta }(n)$ by computing the $k$-th
moments of the average value of $c_{q,\beta }(n)$. In particular we have
provided the first and second moments with improved error terms. We give more
accurate results for the main terms than our predecessors. We also provide an
asymptotic result for an extension of a divisor problem and for an extension of
Ramanujan's formula.

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The categories of this paper are: astro-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: physics
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The categories of this paper are: patt-sol
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The categories of this paper are: cs
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The categories of this paper are: funct-an
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The categories of this paper are: atom-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: math-ph
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The categories of this paper are: stat
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The categories of this paper are: cmp-lg
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The categories of this paper are: adap-org
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The categories of this paper are: nlin
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The categories of this paper are: quant-ph
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The categories of this paper are: econ
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The categories of this paper are: solv-int