My question involves paternity law for the State of: Pennsylvania

The mother's boyfriend who knows he is not the biological father signed an acknowledgement of paternity and the baby was given his last name. A DNA test was done and confirms that he is not the biological father.

The biological father (an exboyfriend) wants his daughter to have his name and wants to be listed as her father on the birth certificate.

The child is almost 4 weeks old.

Does the person who signed the acknowledgement of paternity have any rights? What steps should the biological father take to establish paternity, get his daughter's name changed, and his name on her birth certificate?

The mother has obviously sided with the boyfriend and is against what the biological father wishes to do.