Catalysis of the synthesis of a short RNA primer on a DNA template, providing a free 3'-OH that can be extended by DNA-directed DNA polymerases. In certain conditions, for example in response to DNA damage, some primases synthesize a DNA primer. [ http://www.ncbi.nlm.nih.gov/pubmed/11395402 GOC:mcc http://www.ncbi.nlm.nih.gov/pubmed/38492718 GOC:mah http://www.ncbi.nlm.nih.gov/pubmed/26184436 http://www.ncbi.nlm.nih.gov/pubmed/38203225 ]