Let $ F $ to be a non-archimedean local field, $ chi $ a ramified figure of $ F ^ { ast} $, $ psi $ a non-trivial nature of $ F $, and $ dx $ a measure of Haar on $ F $ regarding the Fourier transformation. The local factor of the Tate epsilon $ epsilon ( chi, psi, dx) $ is defined via the local functional equation

$$ epsilon ( chi, psi, dx) int limits_ {F ^ { ast}} f (x) chi (x) d { ast} x = int limits_ {F ^ { ast}} hat {f} (x) chi ^ {- 1} (x) | x | d ^ { ast} x $$

for all the appropriate functions $ f in C_c (F) $. taking $ f $ to be a characteristic function of an appropriate open compact subgroup of $ F $, we can realize $ epsilon ( chi, psi, dx) $ as "main value integral"

$$ epsilon ( chi, psi, dx) = int limits_ {F} chi ^ {- 1} (x) overline { psi (x)} d { ast} x tag { 1} $$

in that the right side becomes constant and equal to the left side when it is evaluated on $ mathfrak p_F ^ {- k} $ for large enough k $.

Let $ f $ and $ d $ to be the drivers of $ chi $ and $ psi $. The Tate article in Corvallis II describes $ epsilon ( chi, psi, dx) $ more specifically as the right side of (1) where the integral is taken only on the ring of $ x in F ^ { ast} $ with $ operatorname {ord} (x) = -d-f $.

Right here $ operatorname {ord} (c) = d + f $.

My question is, what can we say about the summands

$$ int limit _ { pi ^ n mathcal O ^ { ast}} chi ^ {- 1} (x) psi (x) dx $$

Are they all equal to zero except for $ n = -d-f $? In Tate's thesis, he shows that they are zero for $ n> -d-f $. I wondered if those for $ n <-d-f $ are individually equal to zero, or if they simply cancel each other in the long run.