The Rema, O.C. 583:2 writes that there is a custom not to sleep during the day on Rosh Hashana and that this is a proper custom. This is based on a Yerushalmi which says that one who sleeps on Rosh Hashana his mazel will sleep.
The Bach, O.C. 597:1 states as follows, “The Yerushalmi which says not to sleep on Rosh Hashana is based on the Pasuk, “Uru yeshainim mishinaschem- Awake oh sleepy ones from your slumber”. Meaning how can someone go to sleep while being judged with his life is hanging in the balance? Additionally it shows laziness. However the Maharam MiRutenberg would in fact sleep on Rosh Hashana, like on any other Yom Tov.”