The polynomial X²+Y⁴ captures its primes

1998/11/01 by John Friedlander, Henryk Iwaniec · 1 citation
Mathematics · #math.NT

paper · pdf

published as Ann. of Math. (2) 148 (1998), no. 3, 945-1040 · 96 pages, published version, abstract added in migration

arxiv created 1998/11/01 · arxiv updated 2009/11/30

Abstract

This article proves that there are infinitely many primes of the form a² + b⁴, in fact getting the asymptotic formula. The main result is that ∑_{a² + b⁴≤ x} Λ(a² + b⁴) = 4π^-1κx^3/4 (1 + O(loglog x / log x)) where a, b run over positive integers and κ= ∫¹₀ (1 - t⁴)^1/2 dt = Γ(1/4)² /6√(2π). Here of course, Λdenotes the von Mangoldt function and Γthe Euler gamma function.

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