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It is generally known that
under the generalized Riemann hypothesis one could establish the twin primes
conjecture by the circle method, provided one could obtain the estimate o (nlog-2 n) for the integral of the representation
function over the minor arcs. One of the new results here is that the
assumption of GRH can be removed. We compare this and other such
sufficiency results with similar results for the Goldbach conjecture.