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In this note, assuming a variant of the Generalized Riemann Hypothesis, whichdoes not exclude the existence of real zeros, we prove an asymptotic formulafor the mean value of the representation function for the sum of two primes inarithmetic progressions. This is an improvement of the result of F. R\"uppel in2009, and the generalization of the result of A. Languasco and A. Zaccagniniconcerning the ordinary Goldbach problem in 2012.

A mean value of the representation function for the sum of two primes in arithmetic progressions