The least inert prime in a real quadratic field

Treviño, Enrique

doi:10.1090/S0025-5718-2012-02579-8

The least inert prime in a real quadratic field
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by Enrique Treviño PDF

Math. Comp. 81 (2012), 1777-1797 Request permission

Abstract:

In this paper, we prove that for any positive fundamental discriminant $D > 1596$, there is always at least one prime $p \leq D^{0.45}$ such that the Kronecker symbol $(D/p) = -1$. This improves a result of Granville, Mollin and Williams, where they showed that the least inert prime $p$ in a real quadratic field of discriminant $D > 3705$ is at most $\sqrt {D}/2$. We use a “smoothed” version of the Pólya–Vinogradov inequality, which is very useful for numerically explicit estimates.

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