1. If you say 'Recently I am reading a book named 'Harry Potter'' or 'I am reading 'Harry Potter''. Both 看 and 读 are okay.
2. Originally, in the past days, 看 meant ‘to read with your eyes’; ‘读’ meant 'to read with your mouth'. Because there is a '目' (which means eyes ) in the low part of Chinese character '看', while the left part of ‘读/讀’ is '言' which means 'to speak with your mouth'.
3. Normally, '看书' and '读书' are the same. But when you want to say 'to read something alout with your mouth', you can only use '读书'. The similar uses are '读文章 vs 看文章', '读报纸 vs 看报纸'
4. An exceptional use of '讀書/读书' is '你在哪所学校读书' which means 'Where do you go to school?'