Ohio Revised Code General Provisions § 3.02

(A) When an elective office becomes vacant and is filled by appointment, such appointee shall hold the office until the appointee's successor is elected and qualified;  and such successor shall be elected for the unexpired term, at the first general election for the office which is vacant that occurs more than forty days after the vacancy has occurred;  provided that when the unexpired term ends within one year immediately following the date of such general election, an election to fill such unexpired term shall not be held and the appointment shall be for such unexpired term.

(B) When an elective office becomes vacant and is filled by appointment, the appointing authority shall, immediately but no later than seven days after making the appointment, certify it to the board of elections and to the secretary of state.  The board of elections or, in the case of an appointment to a statewide office, the secretary of state shall issue a certificate of appointment to the appointee.  Certificates of appointment shall be in such form as the secretary of state shall prescribe.

(C) When an elected candidate fails to qualify for the office to which the candidate has been elected, the office shall be filled as in the case of a vacancy.  Until so filled, the incumbent officer shall continue to hold office.  This section does not postpone the time for such election beyond that at which it would have been held had no such vacancy occurred, or affect the official term, or the time for the commencement thereof, of any person elected to such office before the occurrence of such vacancy.