The Multiple Gauss Sums Behind A007789–A007792

Abstract

In 1994 the author (Z.R.) added four integer sequences, A007789 to A007792, to the On-Line Encyclopedia of Integer Sequences. For three decades each of them carried an editorial note saying that nobody knew which problem about circulant matrices and Gauss sums the terms came from. After thirty years, let us reveal the riddle. The terms are normalized multiple Gauss sums: put the determinant of a circulant matrix into the exponent of an additive character, a root of unity exp(2 pi i (.)/n), and average over all vectors of residues, in dimensions three, four, five, and six. We show that each of the four resulting functions of the modulus takes integer values and is multiplicative, and we evaluate it in closed form at every prime power: by a master formula at the primes that do not divide the dimension, by a refinement sorted by valuation at the higher powers of those primes, and by exact laws at the ramified primes, meaning the primes that do divide the dimension. Integrality is what makes the four entries integer sequences at all, and we prove it here by a p-adic stationary-phase descent, which also gives the exact rate at which these sums decay. We give the closed forms sequence by sequence, extend all four sequences with b-files, supply PARI/GP and Mathematica programs, locate the counting content of the theory in the literature on determinants of circulant matrices over finite chain rings, and propose four further sequences of the same family. A final section reduces the whole wild column at degree p^3 (head, middle and tail at once) to one explicit statement about one polynomial. That statement is proved here, out of a closed form for an index function, so the whole column at degree p^3 is a theorem for every odd prime; the shape statement is checked in addition coefficientwise for p up to 23 and the column cellwise for p up to 31. The paper comes in two parts: Part I states every result, marking the remaining conjectures as such, and Part II proves the rest.