[math-fun] article on new partitions results by ken ono and collaborators that are to be revealed tomorrow
not too much detail here, but sounds interesting http://esciencecommons.blogspot.com/2011/01/new-theories-reveal-nature-of-nu... -- Thane Plambeck tplambeck@gmail.com http://counterwave.com/
I was able to download two relevant papers from here: http://www.aimath.org/ Folsom-Kent-Ono paper is 17 pages, Bruner-Ono paper is 3 pages. Based on the abstracts (below) I think the former is the result described in the recent news story. ---- l-ADIC PROPERTIES OF THE PARTITION FUNCTION AMANDA FOLSOM, ZACHARY A. KENT, AND KEN ONO Celebrating the life of A. O. L. Atkin Abstract. Ramanujan’s famous partition congruences modulo powers of 5,7, and 11 imply that certain sequences of partition generating functions tend l-adically to 0. Although these congruences have inspired research in many directions, little is known about the l-adic behavior of these sequences for primes l ≥ 13. We show that these sequences are governed by “fractal” behavior. Modulo any power of a prime l ≥ 5, these sequences of generating functions l-adically converge to linear combinations of at most ⌊l−1⌋−⌊l2−1⌋ many special q-series. For l ∈ {5,7,11} 12 24l we have ⌊ l−1 ⌋−⌊ l2 −1 ⌋ = 0, thereby giving a conceptual explanation of Ramanujan’s congruences. 12 24l We use the general result to reveal the theory of “multiplicative partition congruences” that Atkin anticipated in the 1960s. His results and observations are examples of systematic infinite families of congruences which exist for all powers of primes 13 ≤ l ≤ 31 since ⌊ l−1 ⌋ − ⌊ l2 −1 ⌋ = 1. ---- AN ALGEBRAIC FORMULA FOR THE PARTITION FUNCTION JAN HENDRIK BRUINIER AND KEN ONO Abstract. We derive a formula for the partition function p(n) as a finite sum of algebraic numbers. The summands are discriminant −24n + 1 singular moduli for a special weak Maass form that we describe in terms of Dedekind’s eta-function and Eisenstein series. ---- On Thu, Jan 20, 2011 at 12:53, Thane Plambeck <tplambeck@gmail.com> wrote:
not too much detail here, but sounds interesting
http://esciencecommons.blogspot.com/2011/01/new-theories-reveal-nature-of-nu...
-- Robert Munafo -- mrob.com Follow me at: mrob27.wordpress.com - twitter.com/mrob_27 - youtube.com/user/mrob143 - rilybot.blogspot.com
On 1/20/2011 2:29 PM, Robert Munafo wrote:
I was able to download two relevant papers from here:
Folsom-Kent-Ono paper is 17 pages, Bruner-Ono paper is 3 pages. Based on the abstracts (below) I think the former is the result described in the recent news story.
I just attended Ono's talk kicking off a weekend conference at Emory University on the number theory of partitions. It was a general audience talk, so he didn't give any technical details beyond what was in two abstracts that were quoted in the previous message. He was in fact talking about the results in both papers; the one with Bruinier was given top billing. The most striking thing for me was the claim that their new formula for p(n) is more efficient for calculation than existing methods; Ono suggested that it should be implemented in Sage and the like. Their paper is so short at the moment because it contains only a statement of their results; Ono said it will include the proofs as soon as he's satisfied with how they've been written up. The talk did a good job of entertaining a mixed audience, even including a picture of the Count from Sesame Street. The experts here for the conference seemed to be impressed. A few of them got the rest of the overflow audience to rise for a standing ovation at the end, something I don't recall seeing at a math talk before. -- Fred W. Helenius fredh@ix.netcom.com
participants (3)
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Fred W. Helenius -
Robert Munafo -
Thane Plambeck