Why Bohr got interested in his radius and what it has led to
Resumen: The Bohr radius, named after the Danish mathematician Harald Bohr, is the largest nonnegative number {tex}{\scriptsize r}{/tex} such that
{tex}\[ \sum_{n=0}^{\infty} |a_n| r^n\le \sup_{|z|<1} |\sum_{n=1}^{\infty }a_n z^n|,\]{/tex}
where the sum on the right-hand side represent an arbitrary function analytic and bounded in the open unit disk. It has been known for almost a century that in fact {tex}{\scriptsize{r}}{/tex} {tex}{\scriptsize\!=\frac13\,{.}}{/tex} I will explain why Bohr got interested in his radius and what his ideas have led to. A central theme in the talk will be the remarkable inequality of Bohnenblust and Hille on the coefficients of homogeneous polynomials. Among other things, we will see how this inequality leads to interesting estimates for Dirichlet polynomials.