Why the Riemann hypothesis doesn't imply Goldbach?

The RH solution based on the Hilbert transformed fractional part function provides a connection to the Goldbach conjecture as it links the distribution of zeros of the Riemann Zeta function to the circle method. It basically goes along with a replacement of the log(x)-function by its periodical counterpart log(sin(x) (which is a L(2) function). The latter one plays also a key role in Ramanujan's theory of divergent series. In combination with the proposed fractional Hilbert space framework this leads to a "Goldbach conjecture" analysis on the Schnirelman constant (resp. his concept of (positive) density) under the Riemann Hypothesis. The considered series are related to appropriately defined generalized (Dirichlet) functions. This then is about a distributional asymptotic analysis (VlV).

Goldbach conjecture, Hypothesis, Riemann ..

Refinements of Goldbach's conjecture,and the generalized Riemann hypothesis.
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There are a number of variations on the basic Goldbach conjecture that are often studied as a sort of warm up to the real thing. For example, the Odd Variant postulates that every odd number greater than 7 is the sum of three odd primes. This was proved, but the proof relied on a modification of the Riemann hypothesis - which is as yet another unproven, well-known, challenge.

Pauli graphs, Riemann hypothesis, Goldbach pairs : …

Finally we show that an averaged strong form of Goldbach's conjecture is equivalent to the Generalized Riemann Hypothesis; as well as a similar equivalence to estimates for the number of ways of writing integers as the sum of k primes.
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19/10/2011 · Riemann Hypothesis and Goldbach Conjecture Proof
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Decidability of the Riemann Hypothesis vs

Goldbach's weak conjecture - Wikipedia