What is the ruling on someone who swore secretly, but moved his tongue saying: "I swear by the number of what is in the heavens and the earth that I will not do such and such," then did that? Is expiation for an oath obligatory upon him, and what is it?

An oath is only valid when sworn by one of Allah's names or attributes, according to the Prophet, peace and blessings be upon him, who said: "Whoever takes an oath, let him swear by Allah or remain silent." What was mentioned is not considered an oath, and there is no expiation for it. The speaker should seek Allah's forgiveness and not swear by other than Allah.