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.