In a small saucepan, combine 1 cup of the water with the sugar, raspberries, lemon zest, mint and salt. Bring to a simmer and cook until the sugar has dissolved and the raspberries have broken apart, 2 minutes. Remove from the heat and let cool completely, then strain into a pitcher.
Add the remaining 3 cups of water, the lemon juice, vodka and ice. Stir until well chilled, then serve.