Given a negative, we give a new upper bound on the number of square-free integers ≪X which are represented by some but not all forms of the genus of a primitive, positive-definite binary quadratic form of discriminant D. We also give an analogous upper bound for square-free integers of the form q+a≪X, where q is prime and is fixed. Combined with the-dimensional sieve of Iwaniec, this yields a lower bound on the number of such integers q+a≪X represented by a binary quadratic form of discriminant D, where D is allowed to grow with X as above. An immediate consequence of this, coming from recent work of Bourgain and Fuchs in , is a lower bound on the number of primes which come up as curvatures in a given primitive integer Apollonian circle packing.
|Original language||English (US)|
|Number of pages||49|
|Journal||International Mathematics Research Notices|
|State||Published - Jan 1 2012|
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