When there is no election of probate judge in any district by reason of two or more having an equal and the highest number of votes, or when a new probate district is created and no provision made for the election of a judge thereof, or whenever it is shown to the Governor that a vacancy is about to exist in said office by reason of the resignation of the incumbent to take effect at a future time or by reason of constitutional limitation, or when there is a vacancy in said office, the Governor may issue writs of election directed to the town clerk or clerks or assistant town clerk or clerks within such district, ordering an election to be held on a day named therein, other than a Saturday or Sunday, to fill such vacancy or impending vacancy, and transmit the same to a state marshal. Such state marshal shall forthwith transmit them to such clerk or clerks, who, on receiving the same, shall warn elections to be held on the day appointed in such writs, in the same manner as state elections are warned. Such elections shall be organized and conducted, and the vote shall be declared and returns made, certified, directed, deposited and transmitted, in the same manner as at a state election. The Secretary of the State, Treasurer and Comptroller shall, within thirty days after any such election, count and declare the votes so returned, and notice shall be given to the person declared elected, in the same manner as is provided in the election of probate judges at state elections. The Secretary of the State shall enter the returns in tabular form in books kept by him for that purpose and present a copy of the same, with the name of, and the total number of votes received by, each of the candidates for said office, to the Governor within ten days thereafter. The Probate Court Administrator shall cite a probate judge to act as a judge in the district during any vacancy in said office in accordance with section 45a-120.