As the Ohio CSEA explains,

Quoting
Administrative Paternity Establishment Overview
Paternity establishment refers to the legal determination of being named as the father of a child(ren). The administrative paternity establishment process can be completed at any time up to and including the child's 23rd birthday. Paternity can be established even if one of the parties resides in another county, state or foreign country.
Paternity can be established in the following ways:
Voluntary Acknowledgement refers to the completion of the JFS 07038, also known as the "Acknowledgment of Paternity Affidavit."
This form can be completed at the hospital when your child is born. The voluntary acknowledgment can also be completed at your local heath department or county child support enforcement agency (CSEA). By signing the affidavit, both parties agree they are the biological parents of the child and choose to be named as the legal parents of the child.
Genetic Testing can be requested by the mother, alleged father(s) or the child or child's guardian. Genetic testing may be conducted provided there is not a final determination of paternity, there is uncertainty as to the parentage of the child or if paternity is contested. The CSEA will order all parties to submit to genetic testing and will issue an administrative paternity order based on the results of the genetic test. For paternity to be established through genetic testing, the results must show at least a 99% probability of parentage.
A Court Order of paternity may also be issued through Juvenile Court or Domestic Relations Court.
If your son and the mother signed a Voluntary Ackowledgment of Paternity, either of them could attempt to rescind that AOP within sixty days of its issuance by following the protocol described on the AOP, and doing so would result in the county Child Support Enforcement Agency (CSEA) initiating paternity proceedings.
If the mother went to the CSEA and initiated that process within the 60 day rescission period, the result would be an administrative order that resolved the question of paternity based upon the result of the DNA test they ordered. If your son was named as the legal father, he should have received notice of the proceedings and be instructed to comply with DNA testing. The AOP includes the legal father's address, so he would have received notices at that address. If he did not receive a copy of the final order, assuming one has been issued, odds are he did not have an accurate address on file with the CSEA. He is free to contact the CSEA to inquire about the status of the case and to try to obtain a copy of any order that was issued.