certainly; by all means; absolutely
Emphatic reduplication of 是非, used to strongly express a desire or invitation. Often used when inviting someone or expressing strong personal intention.
ぜひぜひ遊びに来てください。
Please, by all means, come and visit.
その本、ぜひぜひ読んでみたいです。
I'd absolutely love to read that book.
Please do come visit us!
I really want to watch that movie.
ぜひとも is a more formal, written-style intensifier meaning 'without fail', whereas 是非是非 is a warmer, more conversational way to press the same strong request.
必ず stresses certainty or inevitability ('without exception'), while 是非是非 stresses the speaker's strong personal desire or invitation.
Reduplication of 是非 (ぜひ), itself a compound of 是 (right) and 非 (wrong), originally meaning 'right and wrong' and later 'by all means'. The reduplication adds emphasis.