Abstract
The asymptotic distribution of the roots of the congruence ax ≡ b (mod D), 1 ≤ x ≤ D, as D varies, is investigated. Quantitative estimates are obtained by means of exponential sums combined with sieve methods. As an application of the results it is shown that if an additive arithmetic function satisfies f( an + b) − f( cn + d) = O(1) for all positive integers n, ad ≠ bc, then f( n) = O((log n) 3) must hold. This result is apparently the first bound of any kind in such a situation.
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