User blog comment:Razorgirl/Remastered Skyrim (Observations and Thoughts)/@comment-1746153-20161230120023/@comment-1746153-20161231063214

I know I can (or rather should, since this is how it's intended to work) sell it to Calixto. But I'm too lazy to take this extra step.

Breaking into Hjerim -> talking to Viola with the pamphlet you find there -> talking to the mage and asking him about the time of the next murder -> catching Calixto is I think the fastest way to complete the quest. In fact, I haven't tried doing it without picking up the amulet at all - who knows, might work.