Question: Who serves as President of the Senate?

The Vice President of the United States shall be President of the Senate, but shall have no Vote, unless they be equally divided. (US Constitution, Article I, Section 3, Clause 4)

Excepting the duty to receive the tally of electoral votes for President, the only regular responsibility assigned to the office of the Vice President by the Constitutional Convention was to preside over the Senate and to cast tie-breaking votes. Because this seemed to give the Vice President some legislative responsibility, George Mason argued during the Convention that this was a violation of the separation of powers, that "it mixed to much" the executive and legislative powers. But Roger Sherman responded: "If the Vice President were not to be President of the Senate, he would be without employment."
Yet it was agreed that the Vice President to preside over the Senate, and to vote in case of a tie, solved two important problems. First, it allowed that body - at all times - to come to a definitive resolution, because Th President of the Senate would break tie votes. Second, it preserved the equality of the states in the Senate. Should a Senator be chosen to preside over the body, and should that Senator cast the tie-breaking vote, a state would in effect, increase its representation. Joseph Story, Commentaries o the Constitution of the United States.
...Rarely... does the Vice President sit in modern times. The President Pro Tempore of the Senate is the formal substitute, but normally a junior member of the Senate is assigned to sit in the chair.
(Peter W. Schramm, The Heritage Guide to the Constitution, pg 65)