What does the Bible say about the 40-day period after death?

Nothing at all. Some Russians believe that souls of the dead remain at the site of their death for forty days. This is probably based on the fact that Jesus remained on earth for 40 days after his resurrection (Acts 1:3).

Further reading: Bouchard, M. (2004)  Graveyards: Russian ritual and belief pertaining to the dead, Religion, 34(4), 345-362.