What is the function of 甲 in 桂林山水甲天下? What i know: 


甲 is used for enumeration of items or persons in a dialogue, it corresponds to western 1. ,  a) ,  I. ,  X. 
桂林山水 = Landscape of Guilin
天下 = literary expression for whole china or whole world. 


What i do not know: 


甲天下
桂林山水甲天下
桂林山水天下 (would it make sense without 甲 ?)
甲is the first stem of Celestial or Heavenly Stems. We use 甲 as 第一(NO.1) or 顶级(top). Hence, 桂林山水甲天下 means 桂林山水天下第一。

甲天下 means best in the world or No.1 in the world

Anohter example, Lega Serie A is the top football league in Italy, which is translated as 意大利甲级联赛。