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#### The least prime number represented by a binary quadratic form

##### MPS-Authors
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Max Planck Institute for Mathematics, Max Planck Society;

##### External Resource

https://doi.org/10.4171/JEMS/1031
(Publisher version)

##### Supplementary Material (public)
There is no public supplementary material available
##### Citation

Sardari, N. T. (2021). The least prime number represented by a binary quadratic form. Journal of the European Mathematical Society, 23(4), 1161-1223. doi:10.4171/JEMS/1031.

Cite as: http://hdl.handle.net/21.11116/0000-0008-66B4-A
##### Abstract
Let $D<0$ be a fundamental discriminant and $h(D)$ be the class number of $\mathbb{Q}(\sqrt{D})$. 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 $\pi_{D}(X)$ is the number of primes, which split in $\mathbb{Q}(\sqrt{D})$ with norm in the interval $[X,2X].$ We prove that $$\Big(\frac{\pi_D(X)}{\pi(X)}\Big)^2 \ll \frac{R(X,D)}{h(D)}\Big(1+\frac{h(D)}{\pi(X)}\Big),$$ where $\pi(X)$ is the number of primes in the interval $[X,2X]$ and the implicit constant in $\ll$ is independent of $D$ and $X$.