Since Iām farming Killavolt anyway, I thought why not note some stats so I can see if luck actually works? So Iāve kept track of legendary drops. Note that Iāve only kept track of legendary drops in Mayhem 4, I didnāt test anything on lower levels, nor did I care about white/green/blue/purple drops because at this point they donāt matter anyway, do they?
Spoiler: I think Iāll have to agree with ckx00000, at least when it comes to legendary drops from Killavolt.
Iāve kept stats with three different artifacts: one gives ā+12.67 luckā, the second one is a Loaded Dice that āsubstantially increases luckā, and the third one doesnāt have any luck bonus at all. Iāve killed 120 Killavolts and 480 bandits, and opened 120 white chests.
Iāve done 40 runs with each, half on normal M4 and half on TVHM M4. This doesnāt qualify for scientific proof, but it certainly gives a direction. Results:

All results are in the +/-10% range so thereās no significant statistical difference that canāt be attributed to chance. Interestingly, the Loaded Dice which āsubstantially increases luckā performed worst.
One thing worth noting is that during all of that, Iāve had bandits drop a legendary 4 times, and all of them were when using the +12.67 luck artifact. This may be chance, but it may also hint at the luck stat increasing drops from regular enemies, something that may deserve further observation.
Iāve also addressed the normal Mayhem vs. TVHM Mayhem thing. Result:

This actually didnāt surprise me because I always upheld that thereās probably no difference between normal Mayhem and TVHM Mayhem whatsoever. This video came to a similar conclusion. The fight always felt exactly the same and the XP gained from killing Killavolt was always the same too. It reinforces me in my belief that thereās no difference at all: Mayhem is TVHM + modifiers, no matter where you launch it.
Takeaway: I probably wonāt bother with the +XY luck stat anymore and certainly not with the Loaded Dice.