@article{Tanackov2026, 
author = {Ilija Tanackov and Ivan Pavkov and Dejan Ćebić and Željko Stević},
title = {Spectrum of non-trivial zeros, distribution and magnitude of prime numbers},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {1},
pages = {399-419},
keywords = {Zeta functions, resonances, Riemann hypothesis},
url = {https://www.sciopen.com/article/10.3934/math.2026017},
doi = {10.3934/math.2026017},
abstract = {Spectrum RCS(n) = ∑cos(tjlnn), j∈[1, ∞), based on imaginary values of non-trivial zeros of the zeta function ζ(s) = ζ(½±itj) = 0, Im(s) = ±tj, results in "high peaks" (i.e., resonant values of cosine amplitudes at the prime powers n = pk, k∈ℕ in the negative part of the spectrum). The spectrum of the rth root RCS(n1/r) = ∑cos(tjlnn1/r), r∈ℕ exclusively reaches its resonance values in the values of the degree n = pr. Owing to that fact, prime numbers p1 can be separated from prime powers p1/r, r≥2. The spectrum of the dth degree RCS(nd) = ∑cos(tjlnnd), d∈ℕ exclusively reaches its resonant values exclusively in the values of the root p1/d, d≥2. The spectrum of the dth degree "compresses" the number axis. For an arbitrary real interval (a, b), all the resonances pd of all prime numbers from the interval (ad, bd) are contained in the interval (a, b). The imaginary sine spectrum-ISS and the composite spectrum of the RCS and ISS is developed.}
}