论文标题
Friedlander-iwaniec旋转素数的立方类似物
A Cubic analogue of the Friedlander-Iwaniec spin over primes
论文作者
论文摘要
1998年，弗里德兰德（Friedlander）和伊瓦尼克（Iwaniec）证明了$ a^2+b^4 $的无限数量。为了证明这一点，他们通过Jacobi符号定义了高斯整数的旋转，并且证据中的关键要素之一是表明旋转沿高斯素数均等。为了概括这一点，我们通过在Eisenstein Integers上使用Cubic Restue Cargue $ \ Mathbb {Z} [Z} [QM_3] $来定义$ \ mathbb {Z} [ζ_{12}] = \ Mathbb {Z} [ζ_3，i] $的立方体旋转。我们的主要定理说，立方体旋转是按照$ \ mathbb {z} [ζ_{12}] $的主要理想的量子分配。这一点的证据紧随弗里德兰德和伊瓦尼克的路线。在我们案例中，主要的新功能是无限单元组，这意味着我们需要证明整数环上的立方旋转的定义升至理想上定义明确的功能。如果我们考虑Eisenstein Integers上的$ a^2+b^6 $的素数，我们还可以解释如何产生立方体旋转。
In 1998 Friedlander and Iwaniec proved that there are infinitely many primes of the form $a^2+b^4$. To show this they defined the spin of Gaussian integers by the Jacobi symbol, and one of the key ingredients in the proof was to show that the spin becomes equidistributed along Gaussian primes. To generalize this we define the cubic spin of ideals of $\mathbb{Z}[ζ_{12}]=\mathbb{Z}[ζ_3,i]$ by using the cubic residue character on the Eisenstein integers $\mathbb{Z}[ζ_3]$. Our main theorem says that the cubic spin is equidistributed along prime ideals of $\mathbb{Z}[ζ_{12}]$. The proof of this follows closely along the lines of Friedlander and Iwaniec. The main new feature in our case is the infinite unit group, which means that we need to show that the definition of the cubic spin on the ring of integers lifts to a well-defined function on the ideals. We also explain how the cubic spin arises if we consider primes of the form $a^2+b^6$ on the Eisenstein integers.
