Rotation and spectrum of the wave of 7 on the modulo-30 wheel

In our paper on the wave of a prime [3], every prime p ≥ 7 is a wave on the eight vectorsVR = R + 30n of the modulo-30 wheel, and the wave of p multiplies the angles of the helixby p. This note studies the wave of 7, which meets every larger prime q in the semiprime 7q,and asks in what sense these semiprimes rotate with a fixed frequency. First, the action ofa wave on the vectors is a permutation fixed by the group (Z/30Z)× ∼= C2 × C4. The waveof 7 is a rotation of order 4 with the two cycles (V7 V19 V13 V31) and (V11 V17 V29 V23), whichare the classes modulo 3. Second, the weighted count of the semiprimes on the wave of pequals θ(X/p) − θ(p − 1) exactly. By the explicit formula, its normalised deviation carriesevery zeta zero γk as a frequency in τ = ln X, with the amplitude reduced by 1/√p and thephase shifted by γk ln p. For p = 7 and 7q ≤ 7 × 107the twelve strongest spectral peakslie within 0.025 of zeros. The fitted amplitudes of the first 100 zeros are within 0.91–1.10of the prediction (median 1.000), and the phases are within 6.8◦. The tests for p = 11and 13 give the same agreement. The count of all points of the wave, by contrast, has nospectrum. Third, a character χ modulo 30 applied to the vectors multiplies the L-spectrumof the wave by the fixed factor χ(p): +1 for χ3, −1 for χ5, and i, a rotation by 90◦, forthe character of order 4. Fourth, the rotation is measured on three clocks. In the variableq the semiprimes rotate with the fixed frequency 7/30, and all other frequencies a/30 havethe amplitudes |c30(7 − a)|/8 given by Ramanujan sums. In the order of the primes there isno fixed frequency, but consecutive semiprimes share a vector in only 3.6% of cases insteadof 12.5%, the bias of consecutive primes. In ln X the frequencies are the zeros of ζ and ofL(s, χ). The work confirms classical theorems on a new object and makes no claim aboutthe Riemann hypothesis. All numbers are reproduced by the code in the appendix.