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Asif Zaman, Primes represented by positive definite binary quadratic forms, The Quarterly Journal of Mathematics, Volume 69, Issue 4, December 2018, Pages 1353–1386, https://doi.org/10.1093/qmath/hay028
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Abstract
Let f be a primitive positive definite integral binary quadratic form of discriminant and let be the number of primes up to x which are represented by f. We prove several types of upper bounds for within a constant factor of its asymptotic size: unconditional, conditional on the Generalized Riemann Hypothesis (GRH) and for almost all discriminants. The key feature of these estimates is that they hold whenever x exceeds a small power of D and, in some cases, this range of x is essentially best possible. In particular, if f is reduced then this optimal range of x is achieved for almost all discriminants or by assuming GRH. We also exhibit an upper bound for the number of primes represented by f in a short interval and a lower bound for the number of small integers represented by f which have few prime factors.