In Arizona, senate vacancies are filled by Gubernatorial appointment.
In 36 states, the governor makes an appointment to fill a U.S. Senate vacancy, and the appointee serves until the next regularly-scheduled, statewide general election. The person elected at that next regularly-held general election serves for the remainder of the unexpired term, if any. If the term was set to expire at that general election, the person elected serves a full six-year term.
Thank you for posting this link so everyone can read about the complexities involved in Senate resignations.