No. A lich, even when it's still "alive", doesn't have any other magical aura than the aura of spells he may have cast on himself. It wouldn't have any sense that its remains would gain any. Instead, undead have different types of auras that can be detected with Detect Evil and Detect Undead. These spells may be able to detect a lingering aura on lich dust, assuming that it has been harvested recently.
As an inanimate object wouldn't allow detect evil or undead to work on the dust anymore than I would allow it to work on a table or door.
You're right. "Lingering Aura: An evil aura lingers after its original source dissipates (in the case of a spell) or is destroyed (in the case of a creature or magic item). If detect evil is cast and directed at such a location, the spell indicates an aura strength of dim (even weaker than a faint aura). How long the aura lingers at this dim level depends on its original power."So the aura lingers around the location where the creature was, not around its remains.
Now the real question is how much dust is in the lich dust item and how much can a liches body be ground up into because at 400gp a pop one could just die repeatedly and grind up their old lich corpses into lich dust and sell the dust and more than make up for the cost of making the phylactery while simultaneously distributing things that can't effect it in the slightest and thusly never come back to bite them in the butt later.