The polynomial X^2+Y^4 captures its primes

Abstract

This article proves that there are infinitely many primes of the form a^2 + b^4, in fact getting the asymptotic formula. The main result is that \sum_{a^2 + b^4\le x} Λ(a^2 + b^4) = 4π^{-1}κx^{3/4} (1 + O(\log\log x / \log x)) where a, b run over positive integers and κ= \int^1_0 (1 - t^4)^{1/2} dt = Γ(1/4)^2 /6\sqrt{2π}. Here of course, Λdenotes the von Mangoldt function and Γthe Euler gamma function.

96 pages, published version, abstract added in migration

Cited by in corpus (2)

• On the $16$-rank of class groups of $\mathbb{Q}(\sqrt{-2p})$ for primes $p\equiv 1\mathrm{mod}4$
• On Hecke Eigenvalues at Piatetski-Shapiro Primes