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We prove, subject to certain hypotheses, that a positive proportion of the a-points of the Riemann zeta-function and Dirichlet L-functions with primitive characters are simple and discuss corresponding results for other functions in the Selberg class. We also prove an unconditional result of this type for the a-points in fixed strips to the right of the line s = 1/2.doi:10.1090/s0002-9939-2012-11275-4 fatcat:mcjjtkzyunbidegitenqmcoida