Heilbronn sums as Gaussian periods: the period polynomial, the index, and Wieferich congruences (extended version)

Abstract

Heilbronn's exponential sum S_d(c) = Σ_{n≤d} e(c·n^d/d²), whose size has been an open problem since Heath-Brown's first nontrivial bound, is at the same time an algebraic number: 1 + η_c, where η_c is a Gaussian period of the degree-d subfield K_d of Q(ζ_{d²}). We develop the algebraic theory of these sums, reached through the normalized Gauss sums h(d,s) = g(d, d²s²)/σ(s), where g(d,n) = Σ_{x mod n} e(x^d/n), which for an odd prime d take exactly the d values v_j = d·S_d(c), sorted by a discrete logarithm computed by the Fermat quotient. The period polynomial, its discriminant, the index [O_{K_d} : Z[η]] and the norm N(1+η) turn out to be governed at every step by Wieferich congruences p^{d−1} ≡ 1 (mod d²). At composite degrees the values leave the real line and the value set becomes a product of hypocycloid regions, whose count and shape we determine. The paper is in three parts — d an odd prime, odd composite, even — with extensive numerics (all d < 20000). This is the extended version, containing a section on the two discrete logarithms (glog and rlog), a table of notation, and three verification tables omitted from the shortened version submitted for journal publication. Deposited with it are the Wolfram Language package HValues.wl, which computes the value set, its rlog labelling and its boundary at every degree d ≥ 2, and three data files: the extrema of v_j/d^{3/2} and of v_j/(d^{3/2}√log d), and the minimal gaps min_{i≠j}|v_i − v_j|, for all 2261 odd primes d < 20000.