Linear (in)dependence of $\Im(\rho_n)$ and fundamental theorem of arithmetic

Hello,

If I'm not mistaken, globally speaking, Riemann's explicit formula establishes a duality between prime numbers and the non trivial zeroes of the Riemann zeta functions. The imaginary parts of these zeroes are widely believed, assuming RH, to be linearly independent over the rationals. My question is: is a violation of this supposed independence compatible with the fundamental theorem of arithmetic which states that every positive integer factors in a unique fashion into a product of prime numbers? In other words, what would be the consequences of such a violation on the distribution of prime numbers? Thank you in advance.

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