No, like anyone who isn’t watching out for traps caused by bad notation. It’s much easier to copy down numbers than it is to alter them appropriately. If you see “e^(pi i/3)”, what stands out is the 3 in the denominator. Except oops, pi actually only means half a circle, so this is a sixth root of unity, not a third one. Part of why I like to just write zeta_n instead of e^(2pi i/n). Sure, this can be avoided with a bit of thought, but thought shouldn’t be required here; notation that forces you to think about something so trivial, is not good notation.
I’ve certainly used it for that—but I pattern it with dropping the subscript n, when it is clear when there is only one particular root of unity we’re basing off of. I’ve never ever seen zeta used.
No, like anyone who isn’t watching out for traps caused by bad notation. It’s much easier to copy down numbers than it is to alter them appropriately. If you see “e^(pi i/3)”, what stands out is the 3 in the denominator. Except oops, pi actually only means half a circle, so this is a sixth root of unity, not a third one. Part of why I like to just write zeta_n instead of e^(2pi i/n). Sure, this can be avoided with a bit of thought, but thought shouldn’t be required here; notation that forces you to think about something so trivial, is not good notation.
omega_n is the notation I most often run across.
Hm, I’ve generally just seen omega for zeta_3.
I’ve certainly used it for that—but I pattern it with dropping the subscript n, when it is clear when there is only one particular root of unity we’re basing off of. I’ve never ever seen zeta used.