Gross–Koblitz formula

In mathematics the Gross–Koblitz formula, introduced by Gross and Koblitz (1979) expresses a Gauss sum using a product of values of the p-adic gamma function. It is an analog of the Chowla–Selberg formula for the usual gamma function. It implies the Hasse–Davenport relation and generalizes the Stickelberger theorem. Boyarsky (1980) gave another proof of the Gross–Koblitz formula using Dwork's work, and Robert (2001) gave an elementary proof.

Statement

The Gross–Koblitz formula states that the Gauss sum τ can be given in terms of the p-adic gamma function Γ_p by

$\tau_q(r) = -\pi^{s_p(r)}\prod_{i\bmod q}\Gamma_p(r^{(i)}/(q-1))$

where

q is a power p^f of a prime p
r is an integer with 0 ≤ r < q–1
r⁽ⁱ⁾ is the integer whose base p expansion is a cyclic permutation of the first f digits of r.
s_p(r) is the sum of the digits of r in base p
τ is the Gauss sum

$\tau_q(r) = \sum_{a^{q-1}=1}a^{-r}\zeta_\pi^{\text{Tr}(a)}$ where the sum is over roots of 1 in the extension Q_p(π)