Unsigned lease by tenants

My boyfriend and I rented an apartment in August. When I met with the landlord to get they keys and lease, he gave me two copies to sign and send him back a copy. He told me that he would call to give me a fax number to send back the lease the next day after i get my boyfriend's signature. I took it to work the next day but he never called. I never send him the lease because he would never answer his phone. To this day, my boyfriend and I still havent signed the lease. We are living in this apartment for almost six months and are putting up with a 2 year old from upstairs above us, and been wanting to move out. Can we give him 30 day notice without him going after us? Both lease or not signed at all. The landlord would never pick up our calls.