In any case in which an action for a vote by the voters of a district is to be initiated by the petition of such voters, in addition to such other requirements as the general statutes or any special act may impose, such petition shall be on a form prescribed or approved by the clerk of such district, and each page of such petition shall contain a statement, signed under penalties of false statement, by the person who circulated the same, setting forth such circulator’s name and address, and stating that each person whose name appears on said page signed the same in person in the presence of such circulator, that the circulator either knows each such signer or that the signer satisfactorily identified himself to the circulator and that all the signatures on said page were obtained not earlier than six months prior to the filing of said petition. Any page of a petition which does not contain such a statement by the circulator shall be invalid. Any circulator who makes a false statement in the statement hereinbefore provided shall be subject to the penalty provided for false statement. No petition shall be valid for any action for a vote by the voters at any regular or special district meeting unless such petition shall be circulated by a person resident or eligible to vote in such district.