Results on the first moment of cubic Gauss sums and their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Basic identities of cubic Gauss sums — Proposition 3.1. |G(a)| = mu^2(a); G(ab) = conj(chi_b(a)) G(a) G(b) for coprime primary a, b; G(conj a) = conj G(a).
• Kummer's sums — Proposition 3.2. For p = 1 mod 3 and a primary prime pi with pi conj(pi) = p, G(pi) + G(conj pi) = S_p/sqrt(p), so S_p/(2 sqrt p) = Re G(pi).
• The cube of a Gauss sum — Theorem 3.3. G(pi)^3 = -pi/|pi| for every primary prime pi: by Jacobi sums for split primes, by a coset decomposition for inert primes.
• Patterson's first moment — Theorem 4.1. The sum of G(pi) over primary primes of norm at most X is (6/5) c_* X^(5/6)/log X + o(X^(5/6)/log X). Proved in the seventh lesson, using the named results N1 to N3.
• The rational form — Corollary 4.4. The sum of S_p/(2 sqrt p) over primes p <= X is (3/5) c_* X^(5/6)/log X + o(X^(5/6)/log X), half the coefficient of the sum over primary primes.
• The ordinary character large sieve — Lemma 3.3. For primitive characters modulo primary squarefree moduli of norm at most Q and coefficients of length Z, the square sum is at most a constant times (Z + Q^2) times the coefficient norm, with no power of Q lost.
• Extensions of the cubic large sieve — Lemma 4.2. Heath-Brown's cubic large sieve (named result N1) extended to inner coefficients that are not squarefree and to outer variables over all of Z[omega], at the cost of X^epsilon.
• The functional equation for Q(omega) — Proposition 6.2. For a primitive Hecke character of conductor norm D and angular order l, the completion (sqrt(3D)/(2 pi))^s Gamma(s + |l|/2) L(s) satisfies a functional equation with a root number of absolute value 1.
• Type I comparison — Proposition 4.1. For outer length R << X^0.51, the Type I sums of G minus the model, in a fixed angular mode, are O(X^(5/6 - 1/100)). Uses the level Voronoi formula (named result N2) and the cubic large sieve.
• Type I sums at large heights — Proposition 5.2. The mean over heights |t| ~ T of the twisted Type I sums is O(X^(1/2 + epsilon) R^(3/4) T^(1/2)), from the level formula alone.
• Exceptional character moments — Proposition 1.1. For convolutions of smoothly weighted prime sequences, the square sums over coprime squarefree p, q near the configurations (Y, 1) and (Y^(1/3), Y^(1/3)) save a fixed power over Y^(7/3).
• Corrected dispersion — Proposition 1.1. After subtracting the model c_* conj(G(a)) N(a)^(-1/6) S(u), the second moment over a of the bilinear form in b is O(Q L^(-K)); the frequencies that are cubes produce exactly the square of the model.
• The constant of the cube frequencies — Lemma 7.1. For a radial weight W, the integral of the transform of W at z^3 equals c_*^2 times the integral of W(y) |y|^(-2/3).
• Prime decomposition — Proposition 1.1. The prime sum is an exact combination of L^O(1) Type I and bilinear sums with independent coefficients, the same for the Gauss sums and the model; only products of three primes near X^(1/3) escape a power saving.
• The dyadic comparison — Theorem 6.1. For every fixed angular order l, the sum over X < N(pi) <= 2X of theta_l(pi)(G(pi) - c_* N(pi)^(-1/6)) is o(X^(5/6)/log X); with l = 0 this proves Theorem 4.1 of the first lesson.
• Fixed powers — Corollary 2.1. For every fixed k not divisible by 3, the sum of G(pi)^k over N(pi) <= X is the main term of the first moment for k = 1 and k = -1, and o(X^(5/6)/log X) otherwise.

Reading order

Eight lessons, to be read in order. The first sets up the Eisenstein integers, the cubic Gauss sums and the theorem. The second collects the large sieves and the inputs on Hecke L-functions, and the third evaluates Type I sums with the level Voronoi formula. The fourth and fifth control bilinear sums by exceptional character moments and corrected dispersion. The sixth decomposes the prime sum, the seventh assembles the proof, and the eighth treats angular modes and powers. Three published theorems are used without proof in the programme: Heath-Brown's cubic large sieve, the level Voronoi formula of Dunn and Radziwill, and prime number theorems for Hecke characters of small conductor.