Ohio law is pretty clear:
Quote Quoting Ohio Code § 3705.09 - Filing and registration of birth certificate.
(F)(1) If the mother of a child was married at the time of either conception or birth or between conception and birth, the child shall be registered in the surname designated by the mother, and the name of the husband shall be entered on the certificate as the father of the child. The presumption of paternity shall be in accordance with section 3111.03 of the Revised Code.