Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Minimizing GCD sums and applications to non-vanishing of theta functions
  and to Burgess' inequality   In recent years the question of maximizing GCD sums regained interest due to
its firm link with large values of $L$-functions. In the present paper we
initiate the study of minimizing for positive weights~$w$ of normalized $L^1$-
norm the sum $\sum_{m_1 , m_2 \leqslant N}
w({m_1})w({m_2})\frac{(m_1,m_2)}{\sqrt{m_1m_2}} $. We consider as well the
intertwined question of minimizing a weighted version of the usual
multiplicative energy. We give three applications of our results. Firstly we
obtain a logarithmic refinement of Burgess' bound on character sums
$\displaystyle{\sum_{M<n\leqslant M+N}\chi(n)}$ improving previous results of
Kerr, Shparlinski and Yau. Secondly let us denote by $\theta (x,\chi)$ the
theta series associated to a Dirichlet character $\chi$ modulo $p$.
Constructing a suitable mollifier, we improve a result of Louboutin and the
second author and show that, for any $x>0$, there exists at least $ \gg p/(\log
p)^{ \delta+o(1)}$ (with $\delta=1-\frac{1+\log_2 2}{\log 2} \approx 0.08607$)
even characters such that $\theta(x,\chi) \neq 0$. Lastly we obtain lower
bounds on small moments of character sums.

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The categories of this paper are: math-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-ex
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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: cmp-lg
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The categories of this paper are: mtrl-th
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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: gr-qc
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The categories of this paper are: alg-geom
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The categories of this paper are: q-alg
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The categories of this paper are: hep-lat
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The categories of this paper are: econ
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The categories of this paper are: q-fin