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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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up vote 6 down vote accepted

The duality to which you refer does not mean that linear dependence of the zeros is equivalent to linear dependence of primes (or even their logarithms.)

You might look at Odlyzko and te Riele's disproof of the Mertens' conjecture, and the related literature - the truth of Mertens would have implied linear dependence of the zeros. This goes back to Ingham.

EDIT: Since you thought this was helpful, here's an expanded version. The duality between primes and zeros in the Riemann's Explicit Formula does not match up a single zero with a single prime - if it did you could imagine taking linear combinations on both sides to get a result of the kind you asked about.

Instead, the Explicit Formula does almost exactly the opposite. Because a test function on the primes side corresponds to the (more or less) Fourier Transform on the zeros side, the
Heisenberg Uncertainty Principle comes into play. The more you localize on one side, say the zeros, the more spread out the test function becomes on the primes side (and v. versa.)

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