  We study four sums including the Jacquet--Piatetski-Shapiro--Shalika,
Flicker, Bump--Friedberg, and Jacquet--Shalika sums associated to irreducible
cuspidal representations of general linear groups over finite fields. By
computing explicitly, we relate Asai and Bump--Friedberg gamma factors over
finite fields to those over nonarchimedean local fields through level zero
supercuspidal representation. Via Deligne--Kazhdan close field theory, we prove
that exterior square and Bump--Friedberg gamma factors agree with corresponding
Artin gamma factors of their associated tamely ramified representations through
local Langlands correspondence. We also deduce product formulae for Asai,
Bump--Friedberg, and exterior square gamma factors in terms of Gauss sums. By
combining these results, we examine Jacquet--Piatetski-Shapiro--Shalika,
Flicker--Rallis, Jacquet--Shalika, and Friedberg--Jacquet periods and vectors
and their connections to Rankin-Selberg, Asai, exterior square, and
Bump-Friedberg gamma factors, respectively.
