Ohio Rev. Code § 3513.32
When a special election is found necessary to fill a vacancy, the date of the primary election shall be fixed at the same time and in the same manner as that of the election, by the authority calling such special election. The primary election shall be held at least fifteen days prior to the time fixed for such special election. Declaration of candidacy and certificates for such primary shall be filed and fees shall be paid at least ten days before the date for holding such primary election.
A primary election preceding a special election to fill a vacancy in an office shall be eliminated if no valid declaration of candidacy is filed for such office, or if the number of persons filing such declarations of candidacy as candidates of one political party does not exceed the number of candidates which such political party is entitled to nominate for election to such office.
R.C. §3513.32