Vivement

[stu harris]

In internet chat I keep seeing sentences beginning "Vivement...." For example, "Vivement le weekend prochain!"

From the context this obviously means "Roll on next weekend," but why is it expressed as an adverb? Is this used in daily speech, or is it just Internet argot?

[Léonard]

It is used in daily speech really often, not just on the internet. 

Your example is quite tricky because here vivement is not an adverb, but an interjection (I'm not sure if it's the same term in english).

Vivement as an adverb means: quickly, sharply, deeply (La mort de son grand père l'a vivement affectée. / Le peuple s'est vivement rebellé face au dictateur. )

Vivement, as an interjection, is used when you want impatiently something. (Vivement la fin des cours ! / Vivement la retraite ! / Vivement dimanche ;) ! ) 

Vivement is much more used as an interjection than as an adverb, which is more for litterature. 

[stu harris]

Ah yes. Then "Roll on...." is a perfect translation. Merci.

[lamys araktingi]

j'attends (sous entendu) vivement le week end prochain-

you could say "can't wait for the week end"