JournalsjemsVol. 23, No. 4pp. 1161–1223

The least prime number represented by a binary quadratic form

  • Naser Talebizadeh Sardari

    Max-Planck-Institut für Mathematik, Bonn, Germany
The least prime number represented by a binary quadratic form cover

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Abstract

Let D<0D < 0 be a fundamental discriminant and h(D)h(D) be the class number of the imaginary quadratic field Q(D)\mathbb{Q}(\sqrt{D}). Let R(X,D)R(X,D) be the number of the classes of the binary quadratic forms of discriminant DD which represent a prime number in the interval [X,2XX, 2X]. Moreover, assume that πD(X)\pi_D(X) is the number of primes which split in Q(D)\mathbb{Q}(\sqrt{D}) with norm in the interval [X,2XX, 2X]. We prove that

(πD(X)π(X))2R(X,D)h(D)(1+h(D)π(X)),\Big(\frac{\pi_D(X)}{\pi(X)}\Big)^2 \ll \frac{R(X,D)}{h(D)}\Big(1+\frac{h(D)}{\pi(X)}\Big),

where π(X)\pi(X) is the number of the primes is the number of primes in the interval [X,2XX, 2X] and the implicit constant in \ll is independent of DD and XX.

Cite this article

Naser Talebizadeh Sardari, The least prime number represented by a binary quadratic form. J. Eur. Math. Soc. 23 (2020), no. 4, pp. 1161–1223

DOI 10.4171/JEMS/1031