Kummer Sum - Statistical Questions

Statistical Questions

It is known from the general theory of Gauss sums that

|G(χ)| = √p.

In fact the prime decomposition of G(χ) in the cyclotomic field it naturally lies in is known, giving a stronger form. What Kummer was concerned with was the argument

θ_p

of G(χ). Unlike the quadratic case, where the square of the Gauss sum is known and the precise square root was determined by Gauss, here the cube of G(χ) lies in the Eisenstein integers, but its argument is determined by that of the Eisenstein prime dividing p, which splits in that field.

Kummer made a statistical conjecture about θ_p and its distribution modulo 2π (in other words, on the argument of the Kummer sum on the unit circle). For that to make sense, one has to choose between the two possible χ: there is a distinguished choice, in fact, based on the cubic residue symbol. Kummer used available numerical data for p up to 500 (this is described in the 1892 book Theory of Numbers by George B. Mathews). 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. In 1952 John Von Neumann and Herman Goldstine extended Kummer's computations, on ENIAC (written up in John von Neumann and H.H. Goldstine, A Numerical Study of a Conjecture of Kummer 1953).

In the twentieth century, progress was finally made on this question, which had been left untouched for over 100 years. Building on work of Tomio Kubota, S. J. Patterson and Roger Heath-Brown in 1978 proved a modified form of Kummer conjecture. In fact they showed that there was equidistribution of the θ_p. This work involved automorphic forms for the metaplectic group, and Vaughan's lemma in analytic number theory.