Newton's identities of an infinite polynomial with complex-conjugate roots n^−(σ+it) and n^−(σ−it) are multiple zeta functions for n∈[1, ∞), σ∈R and t∈R. All Newton's identities can be represented by Macdonald determinants. In a special case of the Riemann hypothesis, the multiple zeta function of the first order is equal to zero, ζ(σ+it)+ζ(σ−it) = 0. The special case includes all non-trivial zeros. The value of the last, infinite multiple zeta function, in the special case, changes the structure of the determinant that can be calculated. The result is the reciprocal of the factorial value (n!)⁻¹. The general value of the infinite multiple zeta function is calculated based on Vieta's rules and is equal to (n!)^−2σ. The identity based on the relation of the special case and the general case (n!)⁻¹ = (n!)^−2σ is reduced to the equation −1 = −2σ. The value of the critical line for all non-trivial zeros is singular, σ = ½.

Spectrum RCS(n) = ∑cos(t_jlnn), j∈[1, ∞), based on imaginary values of non-trivial zeros of the zeta function ζ(s) = ζ(½±it_j) = 0, Im(s) = ±t_j, results in "high peaks" (i.e., resonant values of cosine amplitudes at the prime powers n = p^k, k∈ℕ in the negative part of the spectrum). The spectrum of the r^th root RCS(n^1/r) = ∑cos(t_jlnn^1/r), r∈ℕ exclusively reaches its resonance values in the values of the degree n = p^r. Owing to that fact, prime numbers p¹ can be separated from prime powers p^1/r, r≥2. The spectrum of the d^th degree RCS(n^d) = ∑cos(t_jlnn^d), d∈ℕ exclusively reaches its resonant values exclusively in the values of the root p^1/d, d≥2. The spectrum of the d^th degree "compresses" the number axis. For an arbitrary real interval (a, b), all the resonances p^d of all prime numbers from the interval (a^d, b^d) are contained in the interval (a, b). The imaginary sine spectrum-ISS and the composite spectrum of the RCS and ISS is developed.