The \( abc\)-conjecture is applied to various questions involving the number of distinct fields \( \mathbb Q(\sqrt{f(n)})\), as one varies over integers \( n\).

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We give an explicit version of Halasz's Theorem, giving an upper bound on the mean value of multiplicative functions in terms of its "distance" from any function of the form \( n^{it}\). We give examples to show our bounds are best possible within a factor of 10.

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We prove various results (including some conjectures of Montgomery and Vaughan) on the distribution of the values of \( L(1,\chi_d)\), including finding complex moments and extreme values.

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