It is customary not to sleep during the day on Rosh Hashanah. Rather, one should engage in Torah study or recite Tehillim. One who is idle, however, is considered as if he slept. The Talmud Yerushalami notes:

If one sleeps at the beginning of the year, i.e., on Rosh Hashanah, his good fortune also sleeps.

One who is very tired to the extent where he will find it difficult to recite Minchah with the proper intention is permitted to take a brief nap in the afternoon.