On August 22, the Arizona Secretary of State posted the official election returns for the August 2, 2022 primary. See them here.
Under legislation passed in 2015, write-in candidates in party primaries need a very large number of write-ins, in order to qualify for the general election ballot. None of the Libertarian candidates for federal or state office polled enough write-ins, which is no surprise, because no minor party has been able to meet those requirements in any of the four primaries held since the new law passed.
For the statewide offices, the Libertarian write-in totals are: Governor Barry Hess 550, Attorney General Michael Kielsky 571, Superintendent of Public Instruction Sheila Reid-Shaver 557, Corporation Commissioner Nathan Gage Madden 216. All of them needed 3,715 write-ins. Of course, only registered Libertarians could cast a write-in vote in the Libertarian primary.
If a candidate is on the primary ballot, there is no minimum number of votes needed. Libertarian U.S. Senate candidate Marc Victor was on the Libertarian primary ballot, so he only needed one vote. In the primary he received 3,065 votes.
The law is more lenient for parties that are on the ballot as a result of a petition that was completed in either of the last two elections. For these “new” parties, only one write-in vote is needed in that party’s primary. But the Libertarian Party is not a “new” party because it has been on the ballot without interruption since 1996.