Reciprocal exponents in the prime factorization: injectivity, algebraic degree, and abscissae of the moments

The multiplicative function J(n), equal to the product of p to the power -1/a over the prime powers p^a exactly dividing n, is injective. The degree of the field Q(J(n)) over Q is the least common multiple of the positive exponents of n. The Dirichlet series of J(n) to the power t has abscissa of absolute convergence equal to the maximum over a of 1/a - t/a^2. On each open interval between the points t_m = m(m+1)/(2m+1), the partial sum is a constant times x to that abscissa, plus an error of the size of the second abscissa. At each transition point the sum gains a factor log x. The thirteen-term summatory expansion down to x to the power 1/16 is left as an open question. There is no claim about prime gaps or the Goldbach conjecture. The record contains the English version (main file), a faithful Portuguese version, and the supplementary archive.Paper 1 of 3 on reciprocal exponents in the prime factorization.