by all means; certainly; definitely
ぜひ expresses a strong desire or recommendation to do something, meaning 'by all means' or 'definitely'.
Use ぜひ when you really want something to happen or when you strongly encourage someone to do something. It adds enthusiasm and sincerity. You can use it with requests, invitations, or expressions of your own desire. For example, before てください it means 'please, by all means', and before たい it means 'I really want to'. It is common in both speaking and writing, but in casual speech it might sound a bit assertive, so it's often softened with ぜひぜひ or used with お願いします. It is not used for negative statements—you don't say ぜひしない.