The least prime number represented by a binary quadratic form

2018/03/08 by Sardari, Naser T.
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1803.03218

Abstract

Let D<0 be a fundamental discriminant and h(D) be the class number of ℚ(√(D)). Let R(X,D) be the number of classes of the binary quadratic forms of discriminant D which represent a prime number in the interval [X,2X]. Moreover, assume that π_D(X) is the number of primes, which split in ℚ(√(D)) with norm in the interval [X,2X]. We prove that ((π_D(X))/(π(X)))² ≪ (R(X,D))/(h(D))(1+(h(D))/(π(X))), where π(X) is the number of primes in the interval [X,2X] and the implicit constant in ≪ is independent of D and X.

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