For every positive integer n, let P(n) denote the largest prime factor of n, with the usual convention that P(1) = 1. For an integer q ≥ 1 and a real number z, we define e_q(z) = e(z/q), where e(z) = exp(2πiz) as usual. In Section 3, we consider the problem of bounding the function e(x; q, a) = #{n ≤ x : P(n) ≡ a (mod q)}. For the case of q fixed, this question has been previously considered by Ivic. However, the approach in [11] apparently does not extend to the case where the modulus q is allowed to grow with the parameter x; this is mainly due to the fact that asymptotic formulas for the number of primes in arithmetic progressions are much less precise for growing moduli than those known for a fixed modulus.
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