讲座提要：

It has long been know that if x is any real number there exists an infinitude of rational numbers p/q which satisfy |x-p/q|<1/q^2. There are two natural questions to ask: one is can the function 1/q^2 on the right be repalced by a smaller function to obtain a sharper inequality? The other is what where happen if we order the reduced fractions, i.e.(p,q)=1. In this talk, I will introduce a few results to answer this two questions.