Values of the Euler phi function not divisible by a prescribed odd prime

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Let phi denote Euler's phi function. For a fixed odd prime we give an asymptotic series expansion in the sense of Poincare for the number Eq(x) of n<=x such that q does not divide phi(n). Thereby we improve on a recent theorem of B.K. Spearman and K.S. Williams [Ark. Mat. 44 (2006), 166-181]. Furthermore we resolve, under the Generalized Riemann Hypothesis, which of two approximations to Eq(x) is asymptotically superior using recent results of Y. Ihara on the Euler-Kronecker constant of a number field.