Journal of Number Theory ( IF 0.6 ) Pub Date : 2021-06-15 , DOI: 10.1016/j.jnt.2021.04.027 Fabrizio Zanello
A famous conjecture of Parkin-Shanks predicts that is odd with density 1/2. Despite the remarkable amount of work of the last several decades, however, even showing this density is positive seems out of reach. In a 2018 paper with Judge, we introduced a different approach and conjectured the ‘‘striking” fact that, if for any the multipartition function has positive odd density, then so does . Similarly, the positive odd density of any with would imply that of .
Our conjecture was shown to be a corollary of an earlier conjecture of the same paper. In this brief note, we provide an unconditional proof of it. An important tool will be Chen's recent breakthrough on a special case of our earlier conjecture.
中文翻译:
从多分函数的正奇数密度推导出p ( n )的正奇数密度:无条件证明
Parkin-Shanks 的一个著名猜想预测, 密度为 1/2 是奇数。然而,尽管过去几十年做了大量的工作,但即使显示这种密度是积极的,似乎也遥不可及。在 2018 年与 Judge 的一篇论文中,我们引入了一种不同的方法并推测了“惊人”的事实,即如果有任何 多分区函数 有正的奇数密度,那么 . 类似地,任意的正奇数密度 和 将意味着 .
我们的猜想被证明是同一篇论文的早期猜想的推论。在这个简短的说明中,我们提供了一个无条件的证明。一个重要的工具将是陈最近对我们早期猜想的一个特例的突破。