Gross–Koblitz Formula

In mathematics, the Gross–Koblitz formula, introduced by 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.
gave another proof of the Gross–Koblitz formula (Boyarski being a pseudonym of Bernard Dwork), and
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

:$\tau_q(r) = -\pi^\prod_\Gamma_p \left(\frac \right)$

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 ''f'' digits of ''r'' by ''i'' positions
* ''s''_''p''(''r'') is the sum of the digits of ''r'' in base ''p''
* $\tau_q(r) = \sum_a^\zeta_\pi^$, where the sum is over roots of 1 in the extension Q_''p''(π)
*π satisfies π^{''p'' – 1} = –''p''
*ζ_π is the ''p''th root of 1 congruent to 1+π mod π²

References

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{{DEFAULTSORT:Gross-Koblitz formula
Theorems in algebraic number theory