Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Kloosterman sums with primes to composite moduli   We obtain a new estimate for Kloosterman sum with primes $p\leqslant X$ to
composite modulo $q$, that is, for the exponential sum of the type \[
\sum\limits_{p\leqslant X,\;p\,\nmid q}\exp{\biggl(\frac{2\pi
i}{q}\bigl(a\overline{p}+bp\bigr)\,\biggr)},\quad (ab,q)=1,\quad
p\overline{p}\equiv 1\pmod{q}, \] which is non-trivial in the case when
$q^{\,3/4+\varepsilon}\leqslant X\ll q^{\,3/2}$. We also apply this estimate to
the proof of solvability of some congruences with inverse prime residues
$\pmod{q}$.

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The categories of this paper are: hep-th
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The categories of this paper are: math-ph
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The categories of this paper are: adap-org
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The categories of this paper are: plasm-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: eess
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The categories of this paper are: ao-sci
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The categories of this paper are: nlin
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The categories of this paper are: hep-ph
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The categories of this paper are: funct-an
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The categories of this paper are: bayes-an
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The categories of this paper are: atom-ph
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The categories of this paper are: chao-dyn
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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: q-fin