Write an friendly, convincing e-mail to my landlord asking whether I can get a dog.
I am writing to you today to request permission to get a dog into the unit. As current, responsible tenants, we'd like to make every effort to assure you we will not be bothered.

We would like to set up a time to discuss the conditions you have to qualify us as pet owners. At that time, we are happy to discuss any financial requirements you may have in order for us to keep a dog. 

We promise to provide appropriate levels of exercise, attention and care for our dog. We will continue to keep a clean apartment, take into consideration our dog's interaction with neighbors and ensure our pet is properly cared for whenever we're unable to take care of it ourselves.