Definition
A Kummer sum is therefore a finite sum
taken over r modulo p, where χ is a Dirichlet character taking values in the cube roots of unity, and where e(x) is the exponential function exp(2πix). Given p of the required form, there are two such characters, together with the trivial character.
The cubic exponential sum K(n,p) defined by
is easily seen to be a linear combination of the Kummer sums. In fact it is 3P where P is one of the Gaussian periods for the subgroup of index 3 in the residues mod p, under multiplication, while the Gauss sums are linear combinations of the P with cube roots of unity as coefficients. However it is the Gauss sum for which the algebraic properties hold. Such cubic exponential sums are also now called Kummer sums.
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