The dome of a prime on the modulo-30 wheel

When the hard semiprimes N = pq (q/p ≤ 3.695) of one vector of the modulo-30 wheelare drawn in increasing order, with their half-difference d = (q − p)/2 against their positionnN , they form domes. We give these domes an exact description and show what they addto the classical pictures of Fermat’s representation N = s2 − d2 and Dirichlet’s hyperbola.Every point is given in closed form by the vectors and rows of its factors. The points withthe same smaller factor p lie on a rising side of slope 15/p, and those with the same largerfactor q on a falling side of slope −15/q. The dome of p rises from its square p2to the apexpq∗ at the edge of the band and falls to the square q∗2; for factors on V7, V13, V17, V23 bothsquares lie on the fifth vector V19. In the plane (s, d) every dome, and every semiprime, isa right isosceles triangle whose feet are exactly the squares of the factors and whose rightangle is the semiprime. Its altitudes give a zigzag path to each foot of length (1 + √2) timesthe side, and the vertices of the two mirror paths keep the sum of the factors fixed whiletheir product falls to N. On the wheel the index of N is the geometric mean of the indices ofthe two feet, up to 2, and their arithmetic mean exceeds it by d2/15. Read along the domes,the weight of a dome transmits every zero ρ of ζ(s) with the transfer function rρ − 1, so adome is a filter on the zeros; read along N, the zeros appear at half their frequency becausethe feet are squares. The muted zeros alone give r = 3.694, and the classification with thewaves of small primes predicts the right foot at the first try in 66% of the domes.