The correct answer is option 4, the Chief Justice of India. When both the offices of the President and the Vice-President are vacant, the Chief Justice of India is entrusted with the responsibility of discharging the functions of the President. This provision is mentioned in Article 65(2) of the Constitution of India.

Let`s examine the other options to understand why they are not the correct answers. Option 1, the Speaker of Lok Sabha, does play a significant role in the functioning of the Parliament but does not have the authority to discharge the functions of the President in this scenario. Option 2, the Deputy Chairman of Rajya Sabha, also does not have the powers to act as the President. Option 3, the Prime Minister, is the head of the government and is responsible for the executive branch, but does not have direct authority to perform the functions of the President when both offices are vacant.

Therefore, the correct answer is option 4, the Chief Justice of India, who assumes the responsibilities of the President in the absence of both the President and Vice-President.