OP is partially right in that if he hasn't been told how to serve notice on the landlord concerning a periodic tenancy then indeed the tenant may not have to serve any notice. The reason being is a tenant is not expected to know the law regarding the service of notices, so must be advised at the appropriate time, and in this case it would be as soon as a SPT came into force.
This should not be confused about any provision within the AST which, if it states what a tenant must do if it becomes a statutory periodic tenancy as that can be ignored.
The advice I give should not be construed as a definitive answer, and is without prejudice or liability. You are advised to consult a specialist solicitor or other person of equal legal standing.