Put the potatoes in a medium sauce pan and cover them with water by one inch. Bring to a boil over high heat and allow to boil until a fork can easily be inserted in them, about 15-20 minutes.
Drain the potatoes in a colander in the sink. In the hot pan that the potatoes were cooking in add in the butter and top with the hot and drained potatoes. Set the pan on a cutting board or on a towel on the counter. Let the hot potatoes melt the butter somewhat.
Add in the salt, lemon juice, lemon zest, and dill. Using a potato masher mash the potatoes until all the ingredients are mixed well and the potatoes are the desired consistency.