We talk about multiplicative character over finite fields, which is  a fundamental topic in analytic number theory. Characters are related to Dirichlet $L$-functions, quadratic residues, primitive roots, and so on. An important and hard problem is to estimate all kinds of  character sums. I will talk about a special average of character sums named as Cesaro means, and show that we can remove the logarithmic factor in the Polya-Vinogradov inequality when taking this average. This is based on the speaker’s recent work.