Minimizing GCD sums and applications to non-vanishing of theta functions and to Burgess' inequality

Abstract

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 ${\sum }_{m_1,m_2⩽N}w(m_1)w(m_2)\frac{(m_1,m_2)}{\sqrt{m_1{m}_2}}$. 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 $\sum _{M<n⩽M+N}\chi (n)$ 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 $\theta (x,\chi )\ne 0$. Lastly we obtain lower bounds on small moments of character sums.

arXiv commentThis paper contains a mistake in the proof of the upper bound of Theorem 1. In a new paper co-authored with Gérald Tenenbaum (see arXiv:1906.12203), the mistake has been corrected adding a local restriction on the integers having a fixed number of prime factors