An inequality for means with applications

We show that an almost trivial inequality for the first and second mean of a random variable can be used to give non-trivial improvements on deep results. As applications we improve on results on lower bounds for the Riemann zeta-function on the critical line, the determinant of a skew-symmetric matrix with entries $\pm 1$, and on the maximal order of an irreducible character of the symmetric group

The exponential sum over squarefree integers

We give an upper bound for the exponential sum over squarefree integers. This establishes a conjecture by Br\"udern and Perelli

The Gauss-Bonnet-Chern mass of conformally flat manifolds

In this paper we show positive mass theorems and Penrose type inequalities for the Gauss-Bonnet-Chern mass, which was introduced recently in \cite{GWW}, for asymptotically flat CF manifolds and its rigidity.Comment: 17 pages, references added, the statement of Prop. 4.6 correcte

Sifted character sums and free quotients of Bianchi groups

We show that the Bianchi group $\mathrm{PSL}_2(\mathcal{O}_d)$, where $\mathcal{O}_d$ is the ring of integers in \Q(\sqrt{d}), $d<0$, has a free quotient of rank $\geq |d|^{1/4-\epsilon}$, as $|d|\to\infty$