Ohio Rev. Code § 302.09: Vacancy in office.

When a vacancy occurs in the board of county commissioners or in the office of county auditor, county treasurer, prosecuting attorney, clerk of the court of common pleas, sheriff, county recorder, county engineer, or coroner more than forty days before the next general election for state and county officers, the vacancy shall be filled as provided for in divisions (A) and (B) of section 305.02 of the Revised Code.