Studiously is the -ly form of studious, and it is used in two ways.

The first follows straight from studious: doing something in the careful, serious way of a person deep in their books. 'He worked studiously all evening.' 'She studiously copied out every word.' Here it means with steady, quiet attention.

The second use is far more common in real English, and students are often surprised by it: studiously means carefully and on purpose, especially when someone is deliberately avoiding something. 'He studiously avoided my eyes.' 'She studiously ignored the question.' The effort here is not the effort of studying — it is the effort of making sure not to do something, and of not showing that you are making an effort. That is why you meet the word again and again next to avoid, ignore, and refuse to look.

There is often a small edge of irony in that second use. The writer is pointing out that all this careful not-noticing is itself very noticeable. If someone studiously ignores you, everyone in the room can tell.

It is a formal and mostly written word. In everyday speech people say 'on purpose', 'deliberately', or 'he made sure not to look at me'. If you meet studiously in a novel or a newspaper, check what follows it: a word of working means the first sense, a word of avoiding means the second.