I live upstairs from my landlord and lately her and I have been having disputes. I have now been receiving calls from friends and family of her's that she gave my number to even after I told her not to give out my phone number. Can I sue my landlord for harassment?

Keep a record of the phone calls - you will need some sort of back up. If necessary call the phone company to get records of incoming calls. I would call the police and tell them that you have been receiving harassing phone calls ( again keeping a record) and say that you believe they are originating from your landlord.

Tell the friends and family that have been calling you not to call you anymore. Send your landlord a written request not to give out your phone number to anyone and save a copy for yourself. If these steps do not solve the problem, then press harassment charges.