Patterson's first moment for cubic Gauss sums

The explanation

Take the prime 7. Cube each whole number from 0 to 6 and keep the remainder after dividing by 7. You get 0, 1, 1, 6, 1, 6 and 6. Now treat each remainder r as an arrow of length 1. It points r sevenths of the way round a circle. Add the seven arrows tip to tail. The total points straight along the horizontal axis and has length about 4.741. This total is Kummer's sum for 7.

Kummer's sum exists for every prime that leaves remainder 1 when divided by 3. Divide it by twice the square root of the prime. The result is the normalized sum, a number between −1 and 1. For 7 it is about 0.896. For 13 it is about 0.253, for 19 about −0.306, and for 31 about 0.921. The values look like random points on a scale from −1 to 1.

In a sense they are random. In 1979, Heath-Brown and Patterson proved that the underlying angles spread evenly round the circle. Yet the tally leans one way. Below 30,000 there are 1,610 such primes. Of their normalized sums, 991 are positive, and the total is about 301. Truly random values of this kind would typically total about 28 in either direction. Kummer had suspected a lean toward positive values in 1846.

Even spread and a lean can both be true, because the lean is small. In 1978, Patterson predicted its size. In the paper's simple model, each prime p tilts the tally by about 0.838 divided by the sixth root of pSource 1. Near 30,000 that tilt is about 0.15. Adding it over the same 1,610 primes gives about 298, close to the observed 301. Up to a limit X, the tilts add up to roughly X to the power 5/6, over the logarithm of X. Heath-Brown proved an upper bound of nearly that size in 2000.

Dunn and Radziwiłł proved a smoothed version of Patterson's prediction in a paper published in 2024. Smoothed means primes near the limit X count only partly. Their proof assumed the Generalized Riemann Hypothesis, or GRH. GRH is a famous unproved statement that would guarantee strong cancellation in sums over primes. The obstacle was to get enough cancellation without it, and with a sharp cutoff at X.

This paper proves the prediction without GRH and with the sharp cutoffSource 2,Source 3. Its main statement uses Eisenstein integers, numbers built from whole numbers and a cube root of 1. Each prime like 7 splits into two such primes, a pair of mirror images. Each gets its own normalized Gauss sum. The pair adds up to Kummer's sum divided by the square root of pSource 4. The paper sums these over all such primes up to size X. The total grows like a fixed constant times X to the power 5/6, over the logarithm of XSource 2. The paper does not improve the general tools for arbitrary prime sumsSource 5. It finds the extra cancellation only in the special prime-built sums that arise hereSource 6.

The theorem is a statement about the long run. At 30,000 its rational-prime formula predicts about 263, not the observed 301. It says the ratio tends to 1, with no rate for the errorSource 7. The paper also weights each value by a fixed power of its prime's angle. Then the lean disappears at this scale. This holds for each fixed nonzero power, not uniformly as the power growsSource 8. It settles part of Patterson's companion prediction about powers of the sumsSource 9. The cases with powers divisible by three stay open. One older paper prints two conflicting constants. The paper says which form it proves and does not claim to settle that conflictSource 10.

How sure is this? A computer proof checker called Lean has checked formal versions of the main result and the angle results. The scope note says they cover all primary Eisenstein primes, with no extra conjectural premise. Each statement is matched against a fixed reference statement by a tool called Comparator. The consequences for powers and for the rational-prime form are not in the scope note. No specialist review has been published.

What it doesn't show. This does not give a rate for the error, or settle the power cases divisible by three.