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.
It is known from the general theory of Gauss sums that In fact the prime decomposition of G(χ) in the cyclotomic field it naturally lies in is known, giving a stronger form.
There was, however, a 'law of small numbers' operating, meaning that Kummer's original conjecture, of a lack of uniform distribution, suffered from a small-number bias.
[6] Next year his subsequent work with Heath-Brown disproving Kummer's conjecture showed that in fact it was equidistributed, but whether the order of the asymptotic was correct remained unknown.
[7] More than 20 years later, Heath-Brown closed on the problem, giving a new sieve method, and conjectured that it could be improved to obtain the predicted order.
[8] In 2021 the problem was demonstrated conditionally on the generalized Riemann hypothesis by Alexander Dunn and Maksym Radziwill, who also showed that the sieve of Heath Brown could not be improved as expected.