Mahlburg Karl
Department of Mathematics, University of Wisconsin, 418 Van Vleck Hall, E B, 480 Lincoln Drive, Madison, WI 53706, USA.
Proc Natl Acad Sci U S A. 2005 Oct 25;102(43):15373-6. doi: 10.1073/pnas.0506702102. Epub 2005 Oct 10.
In 1944, Freeman Dyson conjectured the existence of a "crank" function for partitions that would provide a combinatorial proof of Ramanujan's congruence modulo 11. Forty years later, Andrews and Garvan successfully found such a function and proved the celebrated result that the crank simultaneously "explains" the three Ramanujan congruences modulo 5, 7, and 11. This note announces the proof of a conjecture of Ono, which essentially asserts that the elusive crank satisfies exactly the same types of general congruences as the partition function.
1944年，弗里曼·戴森推测存在一个用于划分的“曲柄”函数，它将为拉马努金模11同余式提供一个组合证明。四十年后，安德鲁斯和加尔文成功找到了这样一个函数，并证明了著名的结果：曲柄同时“解释”了模5、7和11的三个拉马努金同余式。本笔记宣布了小野一个猜想的证明，该猜想本质上断言难以捉摸的曲柄与划分函数满足完全相同类型的一般同余式。
