Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan...
Transcript of Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan...
![Page 1: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/1.jpg)
Ramanujan and asymptotic formulas
Kathrin Bringmann
University of Cologne
This research was supported by the Alfried Krupp Prize
April 19, 2013
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 2: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/2.jpg)
Outline
1. Modular forms
2. Mock modular forms
3. Mixed mock modular forms
4. Non-modular objects
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 3: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/3.jpg)
Outline
1. Modular forms
2. Mock modular forms
3. Mixed mock modular forms
4. Non-modular objects
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 4: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/4.jpg)
Outline
1. Modular forms
2. Mock modular forms
3. Mixed mock modular forms
4. Non-modular objects
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 5: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/5.jpg)
Outline
1. Modular forms
2. Mock modular forms
3. Mixed mock modular forms
4. Non-modular objects
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 6: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/6.jpg)
Outline
1. Modular forms
2. Mock modular forms
3. Mixed mock modular forms
4. Non-modular objects
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 7: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/7.jpg)
Partitions
A partition of a positive integer n is a nonincreasing sequence ofpositive integers whose sum is n.
p(n) := # of partitions of n
Examples:
n = 1
p(1) = 11
n = 2
p(2) = 2
2, 1 + 1
n = 3
p(3) = 33, 2 + 1, 1 + 1 + 1
n = 4
p(4) = 54, 3 + 1, 2 + 2, 2 + 1 + 1, 1 + 1 + 1 + 1
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 8: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/8.jpg)
Partitions
A partition of a positive integer n is a nonincreasing sequence ofpositive integers whose sum is n.
p(n) := # of partitions of n
Examples:
n = 1
p(1) = 11
n = 2
p(2) = 2
2, 1 + 1
n = 3
p(3) = 33, 2 + 1, 1 + 1 + 1
n = 4
p(4) = 54, 3 + 1, 2 + 2, 2 + 1 + 1, 1 + 1 + 1 + 1
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 9: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/9.jpg)
Partitions
A partition of a positive integer n is a nonincreasing sequence ofpositive integers whose sum is n.
p(n) := # of partitions of n
Examples:
n = 1
p(1) = 11
n = 2
p(2) = 2
2, 1 + 1
n = 3
p(3) = 33, 2 + 1, 1 + 1 + 1
n = 4
p(4) = 54, 3 + 1, 2 + 2, 2 + 1 + 1, 1 + 1 + 1 + 1
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 10: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/10.jpg)
Partitions
A partition of a positive integer n is a nonincreasing sequence ofpositive integers whose sum is n.
p(n) := # of partitions of n
Examples:
n = 1
p(1) = 11
n = 2
p(2) = 2
2, 1 + 1
n = 3
p(3) = 33, 2 + 1, 1 + 1 + 1
n = 4
p(4) = 54, 3 + 1, 2 + 2, 2 + 1 + 1, 1 + 1 + 1 + 1
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 11: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/11.jpg)
Partitions
A partition of a positive integer n is a nonincreasing sequence ofpositive integers whose sum is n.
p(n) := # of partitions of n
Examples:
n = 1
p(1) = 1
1
n = 2
p(2) = 2
2, 1 + 1
n = 3
p(3) = 33, 2 + 1, 1 + 1 + 1
n = 4
p(4) = 54, 3 + 1, 2 + 2, 2 + 1 + 1, 1 + 1 + 1 + 1
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 12: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/12.jpg)
Partitions
A partition of a positive integer n is a nonincreasing sequence ofpositive integers whose sum is n.
p(n) := # of partitions of n
Examples:
n = 1 p(1) = 11
n = 2
p(2) = 2
2, 1 + 1
n = 3
p(3) = 33, 2 + 1, 1 + 1 + 1
n = 4
p(4) = 54, 3 + 1, 2 + 2, 2 + 1 + 1, 1 + 1 + 1 + 1
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 13: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/13.jpg)
Partitions
A partition of a positive integer n is a nonincreasing sequence ofpositive integers whose sum is n.
p(n) := # of partitions of n
Examples:
n = 1 p(1) = 11
n = 2
p(2) = 2
2, 1 + 1
n = 3
p(3) = 33, 2 + 1, 1 + 1 + 1
n = 4
p(4) = 54, 3 + 1, 2 + 2, 2 + 1 + 1, 1 + 1 + 1 + 1
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 14: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/14.jpg)
Partitions
A partition of a positive integer n is a nonincreasing sequence ofpositive integers whose sum is n.
p(n) := # of partitions of n
Examples:
n = 1 p(1) = 11
n = 2
p(2) = 2
2,
1 + 1
n = 3
p(3) = 33, 2 + 1, 1 + 1 + 1
n = 4
p(4) = 54, 3 + 1, 2 + 2, 2 + 1 + 1, 1 + 1 + 1 + 1
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 15: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/15.jpg)
Partitions
A partition of a positive integer n is a nonincreasing sequence ofpositive integers whose sum is n.
p(n) := # of partitions of n
Examples:
n = 1 p(1) = 11
n = 2
p(2) = 2
2, 1 + 1
n = 3
p(3) = 33, 2 + 1, 1 + 1 + 1
n = 4
p(4) = 54, 3 + 1, 2 + 2, 2 + 1 + 1, 1 + 1 + 1 + 1
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 16: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/16.jpg)
Partitions
A partition of a positive integer n is a nonincreasing sequence ofpositive integers whose sum is n.
p(n) := # of partitions of n
Examples:
n = 1 p(1) = 11
n = 2 p(2) = 2
2, 1 + 1
n = 3
p(3) = 33, 2 + 1, 1 + 1 + 1
n = 4
p(4) = 54, 3 + 1, 2 + 2, 2 + 1 + 1, 1 + 1 + 1 + 1
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 17: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/17.jpg)
Partitions
A partition of a positive integer n is a nonincreasing sequence ofpositive integers whose sum is n.
p(n) := # of partitions of n
Examples:
n = 1 p(1) = 11
n = 2 p(2) = 2
2, 1 + 1
n = 3
p(3) = 33, 2 + 1, 1 + 1 + 1
n = 4
p(4) = 54, 3 + 1, 2 + 2, 2 + 1 + 1, 1 + 1 + 1 + 1
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 18: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/18.jpg)
Partitions
A partition of a positive integer n is a nonincreasing sequence ofpositive integers whose sum is n.
p(n) := # of partitions of n
Examples:
n = 1 p(1) = 11
n = 2 p(2) = 2
2, 1 + 1
n = 3
p(3) = 3
3,
2 + 1, 1 + 1 + 1
n = 4
p(4) = 54, 3 + 1, 2 + 2, 2 + 1 + 1, 1 + 1 + 1 + 1
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 19: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/19.jpg)
Partitions
A partition of a positive integer n is a nonincreasing sequence ofpositive integers whose sum is n.
p(n) := # of partitions of n
Examples:
n = 1 p(1) = 11
n = 2 p(2) = 2
2, 1 + 1
n = 3
p(3) = 3
3, 2 + 1,
1 + 1 + 1
n = 4
p(4) = 54, 3 + 1, 2 + 2, 2 + 1 + 1, 1 + 1 + 1 + 1
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 20: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/20.jpg)
Partitions
A partition of a positive integer n is a nonincreasing sequence ofpositive integers whose sum is n.
p(n) := # of partitions of n
Examples:
n = 1 p(1) = 11
n = 2 p(2) = 2
2, 1 + 1
n = 3
p(3) = 3
3, 2 + 1, 1 + 1 + 1
n = 4
p(4) = 54, 3 + 1, 2 + 2, 2 + 1 + 1, 1 + 1 + 1 + 1
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 21: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/21.jpg)
Partitions
A partition of a positive integer n is a nonincreasing sequence ofpositive integers whose sum is n.
p(n) := # of partitions of n
Examples:
n = 1 p(1) = 11
n = 2 p(2) = 2
2, 1 + 1
n = 3 p(3) = 33, 2 + 1, 1 + 1 + 1
n = 4
p(4) = 54, 3 + 1, 2 + 2, 2 + 1 + 1, 1 + 1 + 1 + 1
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 22: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/22.jpg)
Partitions
A partition of a positive integer n is a nonincreasing sequence ofpositive integers whose sum is n.
p(n) := # of partitions of n
Examples:
n = 1 p(1) = 11
n = 2 p(2) = 2
2, 1 + 1
n = 3 p(3) = 33, 2 + 1, 1 + 1 + 1
n = 4
p(4) = 54, 3 + 1, 2 + 2, 2 + 1 + 1, 1 + 1 + 1 + 1
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 23: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/23.jpg)
Partitions
A partition of a positive integer n is a nonincreasing sequence ofpositive integers whose sum is n.
p(n) := # of partitions of n
Examples:
n = 1 p(1) = 11
n = 2 p(2) = 2
2, 1 + 1
n = 3 p(3) = 33, 2 + 1, 1 + 1 + 1
n = 4
p(4) = 5
4,
3 + 1, 2 + 2, 2 + 1 + 1, 1 + 1 + 1 + 1
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 24: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/24.jpg)
Partitions
A partition of a positive integer n is a nonincreasing sequence ofpositive integers whose sum is n.
p(n) := # of partitions of n
Examples:
n = 1 p(1) = 11
n = 2 p(2) = 2
2, 1 + 1
n = 3 p(3) = 33, 2 + 1, 1 + 1 + 1
n = 4
p(4) = 5
4, 3 + 1,
2 + 2, 2 + 1 + 1, 1 + 1 + 1 + 1
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 25: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/25.jpg)
Partitions
A partition of a positive integer n is a nonincreasing sequence ofpositive integers whose sum is n.
p(n) := # of partitions of n
Examples:
n = 1 p(1) = 11
n = 2 p(2) = 2
2, 1 + 1
n = 3 p(3) = 33, 2 + 1, 1 + 1 + 1
n = 4
p(4) = 5
4, 3 + 1, 2 + 2,
2 + 1 + 1, 1 + 1 + 1 + 1
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 26: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/26.jpg)
Partitions
A partition of a positive integer n is a nonincreasing sequence ofpositive integers whose sum is n.
p(n) := # of partitions of n
Examples:
n = 1 p(1) = 11
n = 2 p(2) = 2
2, 1 + 1
n = 3 p(3) = 33, 2 + 1, 1 + 1 + 1
n = 4
p(4) = 5
4, 3 + 1, 2 + 2, 2 + 1 + 1,
1 + 1 + 1 + 1
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 27: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/27.jpg)
Partitions
A partition of a positive integer n is a nonincreasing sequence ofpositive integers whose sum is n.
p(n) := # of partitions of n
Examples:
n = 1 p(1) = 11
n = 2 p(2) = 2
2, 1 + 1
n = 3 p(3) = 33, 2 + 1, 1 + 1 + 1
n = 4
p(4) = 5
4, 3 + 1, 2 + 2, 2 + 1 + 1, 1 + 1 + 1 + 1
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 28: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/28.jpg)
Partitions
A partition of a positive integer n is a nonincreasing sequence ofpositive integers whose sum is n.
p(n) := # of partitions of n
Examples:
n = 1 p(1) = 11
n = 2 p(2) = 2
2, 1 + 1
n = 3 p(3) = 33, 2 + 1, 1 + 1 + 1
n = 4 p(4) = 54, 3 + 1, 2 + 2, 2 + 1 + 1, 1 + 1 + 1 + 1
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 29: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/29.jpg)
Fibonacci numbers
Fibonacci numbers:
Fn = Fn−1 + Fn−2
F0 = 0 F1 = 1
First elements:
1 1 2 3 5
but
p(5) = 7F6 = 8
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 30: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/30.jpg)
Fibonacci numbers
Fibonacci numbers:
Fn = Fn−1 + Fn−2
F0 = 0 F1 = 1
First elements:
1 1 2 3 5
but
p(5) = 7F6 = 8
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 31: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/31.jpg)
Fibonacci numbers
Fibonacci numbers:
Fn = Fn−1 + Fn−2
F0 = 0 F1 = 1
First elements:
1 1 2 3 5
but
p(5) = 7F6 = 8
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 32: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/32.jpg)
Fibonacci numbers
Fibonacci numbers:
Fn = Fn−1 + Fn−2
F0 = 0 F1 = 1
First elements:
1 1 2 3 5
but
p(5) = 7F6 = 8
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 33: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/33.jpg)
Generating functions
Euler:
P(q) :=∞∑n=0
p(n)qn =∞∏n=1
1
1− qn
Pentagonal Number Theorem:
∞∏n=1
(1− qn) =∑n∈Z
(−1)nqn(3n−1)
2
Recursion:
p(k) = p(k − 1) + p(k − 2)− p(k − 5)− p(k − 7)
+ p(k − 12) + p(k − 15)− p(k − 22)− . . .
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 34: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/34.jpg)
Generating functions
Euler:
P(q) :=∞∑n=0
p(n)qn =∞∏n=1
1
1− qn
Pentagonal Number Theorem:
∞∏n=1
(1− qn) =∑n∈Z
(−1)nqn(3n−1)
2
Recursion:
p(k) = p(k − 1) + p(k − 2)− p(k − 5)− p(k − 7)
+ p(k − 12) + p(k − 15)− p(k − 22)− . . .
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 35: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/35.jpg)
Generating functions
Euler:
P(q) :=∞∑n=0
p(n)qn =∞∏n=1
1
1− qn
Pentagonal Number Theorem:
∞∏n=1
(1− qn) =∑n∈Z
(−1)nqn(3n−1)
2
Recursion:
p(k) = p(k − 1) + p(k − 2)− p(k − 5)− p(k − 7)
+ p(k − 12) + p(k − 15)− p(k − 22)− . . .
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 36: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/36.jpg)
Modularity
f : H→ C holomorphic is modular of weight k if for all(a bc d
)∈ SL2(Z)
f
(aτ + b
cτ + d
)= (cτ + d)k f (τ)
plus growth condition
Fourier expansion(q = e2πiτ
)f (τ) =
∑n∈Z
a(n)qn
Kathrin Bringmann Ramanujan and asymptotic formulas
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Modularity
f : H→ C holomorphic is modular of weight k if for all(a bc d
)∈ SL2(Z)
weight
f
(aτ + b
cτ + d
)=
↓(cτ + d)k f (τ)
plus growth condition
Fourier expansion(q = e2πiτ
)f (τ) =
∑n∈Z
a(n)qn
Kathrin Bringmann Ramanujan and asymptotic formulas
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Modularity
f : H→ C holomorphic is modular of weight k if for all(a bc d
)∈ SL2(Z)
weight
f
(aτ + b
cτ + d
)=
↓(cτ + d)k f (τ)
plus growth condition
Fourier expansion(q = e2πiτ
)f (τ) =
∑n∈Z
a(n)qn
Kathrin Bringmann Ramanujan and asymptotic formulas
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Modularity
f : H→ C holomorphic is modular of weight k if for all(a bc d
)∈ SL2(Z)
weight
f
(aτ + b
cτ + d
)=
↓(cτ + d)k f (τ)
plus growth condition
Fourier expansion(q = e2πiτ
)f (τ) =
∑n∈Z
a(n)qn
Kathrin Bringmann Ramanujan and asymptotic formulas
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Examples
1) Dedekind η-function
η(τ) := q1
24
∞∏n=1
(1− qn)
Modularity:
η (τ + 1) = eπi12 η(τ),
η
(−1
τ
)=√−iτη(τ).
2) Theta function
Θ(τ) :=∑n∈Z
qn2.
Kathrin Bringmann Ramanujan and asymptotic formulas
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Examples
1) Dedekind η-function
η(τ) := q1
24
∞∏n=1
(1− qn)
Modularity:
η (τ + 1) = eπi12 η(τ),
η
(−1
τ
)=√−iτη(τ).
2) Theta function
Θ(τ) :=∑n∈Z
qn2.
Kathrin Bringmann Ramanujan and asymptotic formulas
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Examples
1) Dedekind η-function
η(τ) := q1
24
∞∏n=1
(1− qn)
Modularity:
η (τ + 1) = eπi12 η(τ),
η
(−1
τ
)=√−iτη(τ).
2) Theta function
Θ(τ) :=∑n∈Z
qn2.
Kathrin Bringmann Ramanujan and asymptotic formulas
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Growth of p(n)
p(10) = 42
p(50) = 204226
p(100) = 190569292
Asymptotic behavior (Hardy–Ramanujan)
p(n) ∼ 1
4√
3neπ√
2n3 (n→∞)
Kathrin Bringmann Ramanujan and asymptotic formulas
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Growth of p(n)
p(10) = 42
p(50) = 204226
p(100) = 190569292
Asymptotic behavior (Hardy–Ramanujan)
p(n) ∼ 1
4√
3neπ√
2n3 (n→∞)
Kathrin Bringmann Ramanujan and asymptotic formulas
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Growth of p(n)
p(10) = 42
p(50) = 204226
p(100) = 190569292
Asymptotic behavior (Hardy–Ramanujan)
p(n) ∼ 1
4√
3neπ√
2n3 (n→∞)
Kathrin Bringmann Ramanujan and asymptotic formulas
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Growth of p(n)
p(10) = 42
p(50) = 204226
p(100) = 190569292
Asymptotic behavior (Hardy–Ramanujan)
p(n) ∼ 1
4√
3neπ√
2n3 (n→∞)
Kathrin Bringmann Ramanujan and asymptotic formulas
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Exact formula
Kloosterman sum:
Ak(n) =∑
h (mod k)∗
ωh,ke−2πihn
k
Bessel function of order α:
Iα(x) :=∞∑
m=0
1
m!Γ(m + α + 1)
(x
2
)2m+α
Rademacher formula:
p(n) =2π
(24n − 1)34
∞∑k=1
Ak(n)
kI 3
2
(π√
24n − 1
6k
)
Kathrin Bringmann Ramanujan and asymptotic formulas
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Exact formula
Kloosterman sum:
some multiplier↓
Ak(n) =∑
h (mod k)∗
ωh,ke−2πihn
k
Bessel function of order α:
Iα(x) :=∞∑
m=0
1
m!Γ(m + α + 1)
(x
2
)2m+α
Rademacher formula:
p(n) =2π
(24n − 1)34
∞∑k=1
Ak(n)
kI 3
2
(π√
24n − 1
6k
)
Kathrin Bringmann Ramanujan and asymptotic formulas
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Exact formula
Kloosterman sum:
some multiplier↓
Ak(n) =∑
h (mod k)∗
ωh,ke−2πihn
k
Bessel function of order α:
Iα(x) :=∞∑
m=0
1
m!Γ(m + α + 1)
(x
2
)2m+α
Rademacher formula:
p(n) =2π
(24n − 1)34
∞∑k=1
Ak(n)
kI 3
2
(π√
24n − 1
6k
)
Kathrin Bringmann Ramanujan and asymptotic formulas
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Exact formula
Kloosterman sum:
some multiplier↓
Ak(n) =∑
h (mod k)∗
ωh,ke−2πihn
k
Bessel function of order α:
Iα(x) :=∞∑
m=0
1
m!Γ(m + α + 1)
(x
2
)2m+α
Rademacher formula:
p(n) =2π
(24n − 1)34
∞∑k=1
Ak(n)
kI 3
2
(π√
24n − 1
6k
)
Kathrin Bringmann Ramanujan and asymptotic formulas
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Outline
1. Modular forms
2. Mock modular forms
3. Mixed mock modular forms
4. Non-modular objects
Kathrin Bringmann Ramanujan and asymptotic formulas
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Ramanujan’s last letter
”I am extremely sorry for notwriting you a single letter up tonow. I recently discovered very
interesting functions which I call“Mock” ϑ-functions. Unlike the“False” ϑ-functions they enterinto mathematics as beautifully
as the theta functions. I amsending you with this letter some
examples.”
Kathrin Bringmann Ramanujan and asymptotic formulas
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Ramanujan’s last letter
”I am extremely sorry for notwriting you a single letter up tonow. I recently discovered very
interesting functions which I call“Mock” ϑ-functions. Unlike the“False” ϑ-functions they enterinto mathematics as beautifully
as the theta functions. I amsending you with this letter some
examples.”
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 54: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/54.jpg)
Ramanujan’s f (q)
Partition function:
P(q) =∑n≥0
qn2
(q; q)2n
with
(a)n = (a; q)n :=n−1∏j=0
(1− aqj
)
Ramanujan’s mock theta function:
f (q) =∑n≥0
qn2
(−q; q)2n
=∑n≥0
α(n)qn.
Kathrin Bringmann Ramanujan and asymptotic formulas
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Ramanujan’s f (q)
Partition function:
P(q) =∑n≥0
qn2
(q; q)2n
with
(a)n = (a; q)n :=n−1∏j=0
(1− aqj
)
Ramanujan’s mock theta function:
f (q) =∑n≥0
qn2
(−q; q)2n
=∑n≥0
α(n)qn.
Kathrin Bringmann Ramanujan and asymptotic formulas
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Ramanujan’s f (q)
Partition function:
P(q) =∑n≥0
qn2
(q; q)2n
with
(a)n = (a; q)n :=n−1∏j=0
(1− aqj
)
Ramanujan’s mock theta function:
f (q) =∑n≥0
qn2
(−q; q)2n
=∑n≥0
α(n)qn.
Kathrin Bringmann Ramanujan and asymptotic formulas
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Dyson’s challenge for the future
”The mock theta-functions giveus tantalizing hints of a grandsynthesis still to be discovered.
Somehow it should be possible tobuild them into a coherentgroup-theoretical structure,
analogous to the structure ofmodular forms which Hecke builtaround the old theta functions ofJacobi. This remains a challenge
for the future. . . ”
Kathrin Bringmann Ramanujan and asymptotic formulas
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Dyson’s challenge for the future
”The mock theta-functions giveus tantalizing hints of a grandsynthesis still to be discovered.
Somehow it should be possible tobuild them into a coherentgroup-theoretical structure,
analogous to the structure ofmodular forms which Hecke builtaround the old theta functions ofJacobi. This remains a challenge
for the future. . . ”
Kathrin Bringmann Ramanujan and asymptotic formulas
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Harmonic Maass forms
Definition:
F : H→ C is a harmonic weak Maass form if it is modular ofweight k and
∆k(F) = 0
with (τ = x + iy)
∆k := −y 2
(∂2
∂x2+
∂2
∂y 2
)+ iky
(∂
∂x+ i
∂
∂y
).
Moreover growth condition.
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Harmonic Maass forms
Definition:
F : H→ C is a harmonic weak Maass form if it is modular ofweight k and
∆k(F) = 0
with (τ = x + iy)
∆k := −y 2
(∂2
∂x2+
∂2
∂y 2
)+ iky
(∂
∂x+ i
∂
∂y
).
Moreover growth condition.
Kathrin Bringmann Ramanujan and asymptotic formulas
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Harmonic Maass forms
Definition:
F : H→ C is a harmonic weak Maass form if it is modular ofweight k and
∆k(F) = 0
with (τ = x + iy)
∆k := −y 2
(∂2
∂x2+
∂2
∂y 2
)+ iky
(∂
∂x+ i
∂
∂y
).
Moreover growth condition.
Kathrin Bringmann Ramanujan and asymptotic formulas
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Examples
I weight 2 Eisenstein series
E2(τ) := E2(τ)− 3
πy
where
E2(τ) := 1− 24∑n≥1
σ(n)qn
with σ(n) :=∑
d |n d .
I Class number generating function
h(τ) :=∑n≥0
n≡0,3 (mod 4)
H(n)qn +i
8√
2π
∫ i∞
−τ
Θ(w)
(−i(τ + w))32
dw
where H(n) is the Hurwitz class number.
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Examples
I weight 2 Eisenstein series
E2(τ) := E2(τ)− 3
πy
where
E2(τ) := 1− 24∑n≥1
σ(n)qn
with σ(n) :=∑
d |n d .
I Class number generating function
h(τ) :=∑n≥0
n≡0,3 (mod 4)
H(n)qn +i
8√
2π
∫ i∞
−τ
Θ(w)
(−i(τ + w))32
dw
where H(n) is the Hurwitz class number.
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Examples
I weight 2 Eisenstein series
E2(τ) := E2(τ)− 3
πy
where
E2(τ) := 1− 24∑n≥1
σ(n)qn
with σ(n) :=∑
d |n d .
I Class number generating function
h(τ) :=∑n≥0
n≡0,3 (mod 4)
H(n)qn +i
8√
2π
∫ i∞
−τ
Θ(w)
(−i(τ + w))32
dw
where H(n) is the Hurwitz class number.
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Natural splitting
F harmonic Maass form
F = F+ + F−
with
F+(τ) :=∑
n�−∞a+(n)qn,
.
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Natural splitting
F harmonic Maass form
F = F+ + F−↑
holomorphicpart
with
F+(τ) :=∑
n�−∞a+(n)qn,
.
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Natural splitting
F harmonic Maass form
F = F+ + F−↑
holomorphicpart
↑non-holomorphic
part
with
F+(τ) :=∑
n�−∞a+(n)qn,
.
Kathrin Bringmann Ramanujan and asymptotic formulas
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Natural splitting
F harmonic Maass form
F = F+ + F−↑
holomorphicpart
↑non-holomorphic
part
with
F+(τ) :=∑
n�−∞a+(n)qn,
F−(τ) :=∑n>0
a−(n)Γ(k − 1; 4π|n|y)qn.
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Natural splitting
F harmonic Maass form
F = F+ + F−↑
holomorphicpart
↑non-holomorphic
part
with
F+(τ) :=∑
n�−∞a+(n)qn,
F−(τ) :=∑n>0
a−(n)Γ(k − 1; 4π|n|y)qn
↑incomplete gamma
function
.
Kathrin Bringmann Ramanujan and asymptotic formulas
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Returning to f (q)
Ramanujan’s claim:
α(n) ∼ (−1)n+1
2√
neπ√
n6 (n→∞)
Andrews-Dragonette Conjecture:
α(n) =π
(24n − 1)14
∑k≥1
(−1)bk+1
2 c
kA2k
(n −
k(1 + (−1)k
)4
)
× I 12
(π√
24n − 1
12n
)Theorem (B-Ono)
The Andrews-Dragonette Conjecture is true.
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Returning to f (q)
Ramanujan’s claim:
α(n) ∼ (−1)n+1
2√
neπ√
n6 (n→∞)
Andrews-Dragonette Conjecture:
α(n) =π
(24n − 1)14
∑k≥1
(−1)bk+1
2 c
kA2k
(n −
k(1 + (−1)k
)4
)
× I 12
(π√
24n − 1
12n
)
Theorem (B-Ono)
The Andrews-Dragonette Conjecture is true.
Kathrin Bringmann Ramanujan and asymptotic formulas
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Returning to f (q)
Ramanujan’s claim:
α(n) ∼ (−1)n+1
2√
neπ√
n6 (n→∞)
Andrews-Dragonette Conjecture:
α(n) =π
(24n − 1)14
∑k≥1
(−1)bk+1
2 c
kA2k
(n −
k(1 + (−1)k
)4
)
× I 12
(π√
24n − 1
12n
)Theorem (B-Ono)
The Andrews-Dragonette Conjecture is true.
Kathrin Bringmann Ramanujan and asymptotic formulas
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Key ingredient 1: (mock) modularity of f (q)
Build the vector-valued function
F (τ) = (F0(τ),F1(τ),F2(τ))T
:=(
q−1
24 f (q), 2q13ω(
q12
), 2q
13ω(−q
12
))T
with
ω(q) :=∑n≥0
q2n2+2n
(q; q2)2n+1
.
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Key ingredient 1: (mock) modularity of f (q)
Build the vector-valued function
F (τ) = (F0(τ),F1(τ),F2(τ))T
:=(
q−1
24 f (q), 2q13ω(
q12
), 2q
13ω(−q
12
))Twith
ω(q) :=∑n≥0
q2n2+2n
(q; q2)2n+1
.
Kathrin Bringmann Ramanujan and asymptotic formulas
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Non-holomorphic part
Period integral:
G (τ) :.
=
∫ i∞
−τ
(g1(z), g0(z),−g2(z))T√−i(τ + z)
dz ,
where
g0(z) :=∑n∈Z
(−1)n(
n +1
3
)e3πi(n+ 1
3 )2z ,
g1(z) := −∑n∈Z
(n +
1
6
)e3πi(n+ 1
6 )2z ,
g2(z) :=∑n∈Z
(n +
1
3
)e3πi(n+ 1
3 )2z .
Kathrin Bringmann Ramanujan and asymptotic formulas
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Non-holomorphic part
Period integral:
G (τ) :.
=
∫ i∞
−τ
(g1(z), g0(z),−g2(z))T√−i(τ + z)
dz ,
where
g0(z) :=∑n∈Z
(−1)n(
n +1
3
)e3πi(n+ 1
3 )2z ,
g1(z) := −∑n∈Z
(n +
1
6
)e3πi(n+ 1
6 )2z ,
g2(z) :=∑n∈Z
(n +
1
3
)e3πi(n+ 1
3 )2z .
Kathrin Bringmann Ramanujan and asymptotic formulas
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Non-holomorphic part
Period integral:
G (τ) :.
=
∫ i∞
−τ
(g1(z), g0(z),−g2(z))T√−i(τ + z)
dz ,
where
g0(z) :=∑n∈Z
(−1)n(
n +1
3
)e3πi(n+ 1
3 )2z ,
g1(z) := −∑n∈Z
(n +
1
6
)e3πi(n+ 1
6 )2z ,
g2(z) :=∑n∈Z
(n +
1
3
)e3πi(n+ 1
3 )2z .
Kathrin Bringmann Ramanujan and asymptotic formulas
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Non-holomorphic part
Period integral:
G (τ) :.
=
∫ i∞
−τ
(g1(z), g0(z),−g2(z))T√−i(τ + z)
dz ,
where
g0(z) :=∑n∈Z
(−1)n(
n +1
3
)e3πi(n+ 1
3 )2z ,
g1(z) := −∑n∈Z
(n +
1
6
)e3πi(n+ 1
6 )2z ,
g2(z) :=∑n∈Z
(n +
1
3
)e3πi(n+ 1
3 )2z .
Kathrin Bringmann Ramanujan and asymptotic formulas
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Non-holomorphic part
Period integral:
G (τ) :.
=
∫ i∞
−τ
(g1(z), g0(z),−g2(z))T√−i(τ + z)
dz ,
where
g0(z) :=∑n∈Z
(−1)n(
n +1
3
)e3πi(n+ 1
3 )2z ,
g1(z) := −∑n∈Z
(n +
1
6
)e3πi(n+ 1
6 )2z ,
g2(z) :=∑n∈Z
(n +
1
3
)e3πi(n+ 1
3 )2z .
Kathrin Bringmann Ramanujan and asymptotic formulas
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Modularity of f (q)
Completion:
F (τ) := F (τ)− G (τ)
Theorem (Zwegers)
We have
F (τ + 1) =
ζ−124 0 00 0 ζ3
0 ζ3 0
F (τ),
F
(−1
τ
)=√−iτ
0 1 01 0 00 0 −1
F (τ),
with ζn := e2πin . Moreover ∆ 1
2
(F)
= 0.
Kathrin Bringmann Ramanujan and asymptotic formulas
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Modularity of f (q)
Completion:
F (τ) := F (τ)− G (τ)
Theorem (Zwegers)
We have
F (τ + 1) =
ζ−124 0 00 0 ζ3
0 ζ3 0
F (τ),
F
(−1
τ
)=√−iτ
0 1 01 0 00 0 −1
F (τ),
with ζn := e2πin .
Moreover ∆ 12
(F)
= 0.
Kathrin Bringmann Ramanujan and asymptotic formulas
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Modularity of f (q)
Completion:
F (τ) := F (τ)− G (τ)
Theorem (Zwegers)
We have
F (τ + 1) =
ζ−124 0 00 0 ζ3
0 ζ3 0
F (τ),
F
(−1
τ
)=√−iτ
0 1 01 0 00 0 −1
F (τ),
with ζn := e2πin . Moreover ∆ 1
2
(F)
= 0.
Kathrin Bringmann Ramanujan and asymptotic formulas
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Key ingredient 2: Maass Poincare series
General shape:
∑M=
(a bc d
)∈Γ∞\SL2(Z)
(cτ + d)−kφ
(aτ + b
cτ + d
)
with
Γ∞ :=
{±(
1 n0 1
); n ∈ Z
}.
Kathrin Bringmann Ramanujan and asymptotic formulas
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Key ingredient 2: Maass Poincare series
General shape:
∑M=
(a bc d
)∈Γ∞\SL2(Z)
(cτ + d)−kφ
(aτ + b
cτ + d
)
with
Γ∞ :=
{±(
1 n0 1
); n ∈ Z
}.
Kathrin Bringmann Ramanujan and asymptotic formulas
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Maass-Poincare series
Average
ϕs(τ) :=Ms
(−πy
6
)e−
πix12
with
Poincare series (Re(s) > 1):
P 12
(s; τ) :=∑
M∈Γ∞\Γ0(2)
χ(M)−1(cτ + d)−12ϕs(Mτ).
↑χ appropriate
multiplier
Kathrin Bringmann Ramanujan and asymptotic formulas
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Maass-Poincare series
Average
ϕs(τ) :=Ms
(−πy
6
)e−
πix12
with
Ms(u) := |u|−12 M 1
4sgn(u),s− 1
2(|u|)
Poincare series (Re(s) > 1):
P 12
(s; τ) :=∑
M∈Γ∞\Γ0(2)
χ(M)−1(cτ + d)−12ϕs(Mτ).
↑χ appropriate
multiplier
Kathrin Bringmann Ramanujan and asymptotic formulas
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Maass-Poincare series
Average
ϕs(τ) :=Ms
(−πy
6
)e−
πix12
with
Ms(u) := |u|−12 M 1
4sgn(u),s− 1
2(|u|)
↑M-Whittaker
function
Poincare series (Re(s) > 1):
P 12
(s; τ) :=∑
M∈Γ∞\Γ0(2)
χ(M)−1(cτ + d)−12ϕs(Mτ).
↑χ appropriate
multiplier
Kathrin Bringmann Ramanujan and asymptotic formulas
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Maass-Poincare series
Average
ϕs(τ) :=Ms
(−πy
6
)e−
πix12
with
Ms(u) := |u|−12 M 1
4sgn(u),s− 1
2(|u|)
↑M-Whittaker
function
Poincare series (Re(s) > 1):
P 12
(s; τ) :=∑
M∈Γ∞\Γ0(2)
χ(M)−1(cτ + d)−12ϕs(Mτ).
↑χ appropriate
multiplier
Kathrin Bringmann Ramanujan and asymptotic formulas
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Maass-Poincare series
Average
ϕs(τ) :=Ms
(−πy
6
)e−
πix12
with
Ms(u) := |u|−12 M 1
4sgn(u),s− 1
2(|u|)
↑M-Whittaker
function
Poincare series (Re(s) > 1):
P 12
(s; τ) :=∑
M∈Γ∞\Γ0(2)
χ(M)−1(cτ + d)−12ϕs(Mτ).
↑χ appropriate
multiplier
Kathrin Bringmann Ramanujan and asymptotic formulas
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Sketch of the proof
Idea:
I Analytically continue to s = 34
(via Fourier expansion)
I Show
F0(τ) = P 12
(3
4; τ
)
where
F0(τ) := F0(24τ)− G0(24τ).
Kathrin Bringmann Ramanujan and asymptotic formulas
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Sketch of the proof
Idea:
I Analytically continue to s = 34
(via Fourier expansion)
I Show
F0(τ) = P 12
(3
4; τ
)
where
F0(τ) := F0(24τ)− G0(24τ).
Kathrin Bringmann Ramanujan and asymptotic formulas
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Sketch of the proof
Idea:
I Analytically continue to s = 34
(via Fourier expansion)
I Show
F0(τ) = P 12
(3
4; τ
)
where
F0(τ) := F0(24τ)− G0(24τ).
Kathrin Bringmann Ramanujan and asymptotic formulas
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Sketch of the proof
Idea:
I Analytically continue to s = 34
(via Fourier expansion)
I Show
F0(τ) = P 12
(3
4; τ
)where
F0(τ) := F0(24τ)− G0(24τ).
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1. Modular forms
2. Mock modular forms
3. Mixed mock modular forms
4. Non-modular objects
Kathrin Bringmann Ramanujan and asymptotic formulas
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Combinatorics
Joint with K. Mahlburg
Definition:
A partition without sequences is a partition with no adjacent parts
s(n) := # partitions of n without sequences
Example:
6 + 6 + 4 + 1 + 1 + 1 is a partition without sequences of 19
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Combinatorics
Joint with K. Mahlburg
Definition:
A partition without sequences is a partition with no adjacent parts
s(n) := # partitions of n without sequences
Example:
6 + 6 + 4 + 1 + 1 + 1 is a partition without sequences of 19
Kathrin Bringmann Ramanujan and asymptotic formulas
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Combinatorics
Joint with K. Mahlburg
Definition:
A partition without sequences is a partition with no adjacent parts
s(n) := # partitions of n without sequences
Example:
6 + 6 + 4 + 1 + 1 + 1 is a partition without sequences of 19
Kathrin Bringmann Ramanujan and asymptotic formulas
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Combinatorics
Joint with K. Mahlburg
Definition:
A partition without sequences is a partition with no adjacent parts
s(n) := # partitions of n without sequences
Example:
6 + 6 + 4 + 1 + 1 + 1 is a partition without sequences of 19
Kathrin Bringmann Ramanujan and asymptotic formulas
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Generating function
Andrews:
∑n≥0
s(n)qn =
(−q3; q3
)∞
(q2; q2)∞χ(q)
with
χ(q) :=∑n≥0
(−q; q)n(−q3; q3)n
qn2
Kathrin Bringmann Ramanujan and asymptotic formulas
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Generating function
Andrews:
∑n≥0
s(n)qn =
(−q3; q3
)∞
(q2; q2)∞χ(q)
↑modular
with
χ(q) :=∑n≥0
(−q; q)n(−q3; q3)n
qn2
Kathrin Bringmann Ramanujan and asymptotic formulas
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Generating function
Andrews:
∑n≥0
s(n)qn =
(−q3; q3
)∞
(q2; q2)∞χ(q)
↑modular
↑mock
with
χ(q) :=∑n≥0
(−q; q)n(−q3; q3)n
qn2
Kathrin Bringmann Ramanujan and asymptotic formulas
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Algebraic geometry
Joint with J. Manschot
Generating function for Euler numbers:
For j ∈ {0, 1}, we have
fj(τ) =∞∑n=0
αj(n)qn :=
Kathrin Bringmann Ramanujan and asymptotic formulas
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Algebraic geometry
Joint with J. Manschot
Generating function for Euler numbers:
For j ∈ {0, 1}, we have
fj(τ) =∞∑n=0
αj(n)qn :=1
η6(τ)
∞∑n=0
H(4n + 3j)qn+ 3j4
Kathrin Bringmann Ramanujan and asymptotic formulas
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Algebraic geometry
Joint with J. Manschot
Generating function for Euler numbers:
For j ∈ {0, 1}, we have
fj(τ) =∞∑n=0
αj(n)qn :=1
η6(τ)
∞∑n=0
H(4n + 3j)qn+ 3j4
↑modular
Kathrin Bringmann Ramanujan and asymptotic formulas
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Algebraic geometry
Joint with J. Manschot
Generating function for Euler numbers:
For j ∈ {0, 1}, we have
fj(τ) =∞∑n=0
αj(n)qn :=1
η6(τ)
∞∑n=0
H(4n + 3j)qn+ 3j4
↑modular
↑mock
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Lie superalgebras
Joint with K. Ono, K. Mahlburg
Character formula for s`(m|1)∧-modules (s ∈ Z)
2q−s2
(q2; q2
)2
∞(q; q)m+2
∞
∑k=(k1,...,km−1)∈Zm−1
q12
∑m−1i=1 ki (ki+1)
1 + q∑m−1
i=1 ki−s
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Lie superalgebras
Joint with K. Ono, K. Mahlburg
Character formula for s`(m|1)∧-modules (s ∈ Z)
2q−s2
(q2; q2
)2
∞(q; q)m+2
∞
∑k=(k1,...,km−1)∈Zm−1
q12
∑m−1i=1 ki (ki+1)
1 + q∑m−1
i=1 ki−s
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An exact formula
Recall generating function for Euler numbers:
fj(τ) =∞∑n=0
αj(n)qn
Notation:
For k ∈ N, g ∈ Z, u ∈ R
fk,g (u) :=
π2
sinh2(πuk −πig2k )
if g 6≡ 0 (mod 2k),
π2
sinh2(πuk )− k2
u2 if g ≡ 0 (mod 2k).
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An exact formula
Recall generating function for Euler numbers:
fj(τ) =∞∑n=0
αj(n)qn
Notation:
For k ∈ N, g ∈ Z, u ∈ R
fk,g (u) :=
π2
sinh2(πuk −πig2k )
if g 6≡ 0 (mod 2k),
π2
sinh2(πuk )− k2
u2 if g ≡ 0 (mod 2k).
Kathrin Bringmann Ramanujan and asymptotic formulas
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An exact formula (cont.)
Kloosterman sums:
Kj ,` (n,m; k) :=∑
0≤h<k(h,k)=1
ψj`
(h, h′, k
)e− 2πi
k
(hn+ h′n
4
)
Bessel function integral:
Ik,g (n) :=
∫ 1
−1fk,g
(u
2
)I 7
2
(π
k
√(4n − (j + 1)) (1− u2)
)×(1− u2
) 74 du
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An exact formula (cont.)
Kloosterman sums:
Kj ,` (n,m; k) :=∑
0≤h<k(h,k)=1
ψj`
(h, h′, k
)↑
multiplier
e− 2πi
k
(hn+ h′n
4
)
Bessel function integral:
Ik,g (n) :=
∫ 1
−1fk,g
(u
2
)I 7
2
(π
k
√(4n − (j + 1)) (1− u2)
)×(1− u2
) 74 du
Kathrin Bringmann Ramanujan and asymptotic formulas
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An exact formula (cont.)
Kloosterman sums:
Kj ,` (n,m; k) :=∑
0≤h<k(h,k)=1
ψj`
(h, h′, k
)↑
multiplier
e− 2πi
k
(hn+ h′n
4
)
Bessel function integral:
Ik,g (n) :=
∫ 1
−1fk,g
(u
2
)I 7
2
(π
k
√(4n − (j + 1)) (1− u2)
)×(1− u2
) 74 du
Kathrin Bringmann Ramanujan and asymptotic formulas
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An exact formula (cont.)
Theorem (B-Manschot)
The coefficients αj(n) equal
− π
6 (4n − (j + 1))54
∑k≥1
Kj ,0 (n, 0; k)
kI 5
2
(πk
√4n − (j + 1)
)
+1
√2 (4n − (j + 1))
32
∑k≥1
Kj ,0 (n, 0; k)√k
I3(π
k
√4n − (j + 1)
)
− 1
8π (4n − (j + 1))74
∑k≥1
∑`∈{0,1}−k<g≤k
g≡` (mod 2)
Kj ,`
(n, g 2; k
)k2
Ik,g (n).
Kathrin Bringmann Ramanujan and asymptotic formulas
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An exact formula (cont.)
Theorem (B-Manschot)
The coefficients αj(n) equal
− π
6 (4n − (j + 1))54
∑k≥1
Kj ,0 (n, 0; k)
kI 5
2
(πk
√4n − (j + 1)
)
+1
√2 (4n − (j + 1))
32
∑k≥1
Kj ,0 (n, 0; k)√k
I3(π
k
√4n − (j + 1)
)
− 1
8π (4n − (j + 1))74
∑k≥1
∑`∈{0,1}−k<g≤k
g≡` (mod 2)
Kj ,`
(n, g 2; k
)k2
Ik,g (n).
Kathrin Bringmann Ramanujan and asymptotic formulas
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An exact formula (cont.)
Theorem (B-Manschot)
The coefficients αj(n) equal
− π
6 (4n − (j + 1))54
∑k≥1
Kj ,0 (n, 0; k)
kI 5
2
(πk
√4n − (j + 1)
)
+1
√2 (4n − (j + 1))
32
∑k≥1
Kj ,0 (n, 0; k)√k
I3(π
k
√4n − (j + 1)
)
− 1
8π (4n − (j + 1))74
∑k≥1
∑`∈{0,1}−k<g≤k
g≡` (mod 2)
Kj ,`
(n, g 2; k
)k2
Ik,g (n).
Kathrin Bringmann Ramanujan and asymptotic formulas
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An exact formula (cont.)
Theorem (B-Manschot)
The coefficients αj(n) equal
− π
6 (4n − (j + 1))54
∑k≥1
Kj ,0 (n, 0; k)
kI 5
2
(πk
√4n − (j + 1)
)
+1
√2 (4n − (j + 1))
32
∑k≥1
Kj ,0 (n, 0; k)√k
I3(π
k
√4n − (j + 1)
)
− 1
8π (4n − (j + 1))74
∑k≥1
∑`∈{0,1}−k<g≤k
g≡` (mod 2)
Kj ,`
(n, g 2; k
)k2
Ik,g (n).
Kathrin Bringmann Ramanujan and asymptotic formulas
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Main term
Corollary
We have as n→∞
αj(n) =
(1
96n−
32 − 1
32πn−
74 + O
(n−2))
e2π√n.
Kathrin Bringmann Ramanujan and asymptotic formulas
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Main term
Corollary
We have as n→∞
αj(n) =
(1
96n−
32 − 1
32πn−
74 + O
(n−2))
e2π√n.
Kathrin Bringmann Ramanujan and asymptotic formulas
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Outline
1. Modular forms
2. Mock modular forms
3. Mixed mock modular forms
4. Non-modular objects
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Stacks
Definition:
A stack of size n ∈ N0 is a decomposition of n into positive integers
n = a1 + · · ·+ ar + c + bs + · · ·+ b1
such that
0 < a1 ≤ a2 ≤ · · · ≤ ar ≤ c < bs ≥ · · · ≥ b1 > 0.
s(n) := # stacks of size n
Example:
n = 4:4 3+1 1+3 2+2
2+1+1 1+2+1 1+1+2 1+1+1+1
Thus s(4) = 8.
Kathrin Bringmann Ramanujan and asymptotic formulas
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Stacks
Definition:
A stack of size n ∈ N0 is a decomposition of n into positive integers
n = a1 + · · ·+ ar + c + bs + · · ·+ b1
such that
0 < a1 ≤ a2 ≤ · · · ≤ ar ≤ c < bs ≥ · · · ≥ b1 > 0.
s(n) := # stacks of size n
Example:
n = 4:4 3+1 1+3 2+2
2+1+1 1+2+1 1+1+2 1+1+1+1
Thus s(4) = 8.
Kathrin Bringmann Ramanujan and asymptotic formulas
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Stacks
Definition:
A stack of size n ∈ N0 is a decomposition of n into positive integers
n = a1 + · · ·+ ar + c + bs + · · ·+ b1
such that
0 < a1 ≤ a2 ≤ · · · ≤ ar ≤ c < bs ≥ · · · ≥ b1 > 0.
s(n) := # stacks of size n
Example:
n = 4:4 3+1 1+3 2+2
2+1+1 1+2+1 1+1+2 1+1+1+1
Thus s(4) = 8.
Kathrin Bringmann Ramanujan and asymptotic formulas
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Stacks
Definition:
A stack of size n ∈ N0 is a decomposition of n into positive integers
n = a1 + · · ·+ ar + c + bs + · · ·+ b1
such that
0 < a1 ≤ a2 ≤ · · · ≤ ar ≤ c < bs ≥ · · · ≥ b1 > 0.
s(n) := # stacks of size n
Example:
n = 4:
4 3+1 1+3 2+2
2+1+1 1+2+1 1+1+2 1+1+1+1
Thus s(4) = 8.
Kathrin Bringmann Ramanujan and asymptotic formulas
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Stacks
Definition:
A stack of size n ∈ N0 is a decomposition of n into positive integers
n = a1 + · · ·+ ar + c + bs + · · ·+ b1
such that
0 < a1 ≤ a2 ≤ · · · ≤ ar ≤ c < bs ≥ · · · ≥ b1 > 0.
s(n) := # stacks of size n
Example:
n = 4:4
3+1 1+3 2+2
2+1+1 1+2+1 1+1+2 1+1+1+1
Thus s(4) = 8.
Kathrin Bringmann Ramanujan and asymptotic formulas
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Stacks
Definition:
A stack of size n ∈ N0 is a decomposition of n into positive integers
n = a1 + · · ·+ ar + c + bs + · · ·+ b1
such that
0 < a1 ≤ a2 ≤ · · · ≤ ar ≤ c < bs ≥ · · · ≥ b1 > 0.
s(n) := # stacks of size n
Example:
n = 4:4 3+1
1+3 2+2
2+1+1 1+2+1 1+1+2 1+1+1+1
Thus s(4) = 8.
Kathrin Bringmann Ramanujan and asymptotic formulas
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Stacks
Definition:
A stack of size n ∈ N0 is a decomposition of n into positive integers
n = a1 + · · ·+ ar + c + bs + · · ·+ b1
such that
0 < a1 ≤ a2 ≤ · · · ≤ ar ≤ c < bs ≥ · · · ≥ b1 > 0.
s(n) := # stacks of size n
Example:
n = 4:4 3+1 1+3
2+2
2+1+1 1+2+1 1+1+2 1+1+1+1
Thus s(4) = 8.
Kathrin Bringmann Ramanujan and asymptotic formulas
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Stacks
Definition:
A stack of size n ∈ N0 is a decomposition of n into positive integers
n = a1 + · · ·+ ar + c + bs + · · ·+ b1
such that
0 < a1 ≤ a2 ≤ · · · ≤ ar ≤ c < bs ≥ · · · ≥ b1 > 0.
s(n) := # stacks of size n
Example:
n = 4:4 3+1 1+3 2+2
2+1+1 1+2+1 1+1+2 1+1+1+1
Thus s(4) = 8.
Kathrin Bringmann Ramanujan and asymptotic formulas
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Stacks
Definition:
A stack of size n ∈ N0 is a decomposition of n into positive integers
n = a1 + · · ·+ ar + c + bs + · · ·+ b1
such that
0 < a1 ≤ a2 ≤ · · · ≤ ar ≤ c < bs ≥ · · · ≥ b1 > 0.
s(n) := # stacks of size n
Example:
n = 4:4 3+1 1+3 2+2
2+1+1
1+2+1 1+1+2 1+1+1+1
Thus s(4) = 8.
Kathrin Bringmann Ramanujan and asymptotic formulas
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Stacks
Definition:
A stack of size n ∈ N0 is a decomposition of n into positive integers
n = a1 + · · ·+ ar + c + bs + · · ·+ b1
such that
0 < a1 ≤ a2 ≤ · · · ≤ ar ≤ c < bs ≥ · · · ≥ b1 > 0.
s(n) := # stacks of size n
Example:
n = 4:4 3+1 1+3 2+2
2+1+1 1+2+1
1+1+2 1+1+1+1
Thus s(4) = 8.
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 130: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/130.jpg)
Stacks
Definition:
A stack of size n ∈ N0 is a decomposition of n into positive integers
n = a1 + · · ·+ ar + c + bs + · · ·+ b1
such that
0 < a1 ≤ a2 ≤ · · · ≤ ar ≤ c < bs ≥ · · · ≥ b1 > 0.
s(n) := # stacks of size n
Example:
n = 4:4 3+1 1+3 2+2
2+1+1 1+2+1 1+1+2
1+1+1+1
Thus s(4) = 8.
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 131: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/131.jpg)
Stacks
Definition:
A stack of size n ∈ N0 is a decomposition of n into positive integers
n = a1 + · · ·+ ar + c + bs + · · ·+ b1
such that
0 < a1 ≤ a2 ≤ · · · ≤ ar ≤ c < bs ≥ · · · ≥ b1 > 0.
s(n) := # stacks of size n
Example:
n = 4:4 3+1 1+3 2+2
2+1+1 1+2+1 1+1+2 1+1+1+1
Thus s(4) = 8.
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 132: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/132.jpg)
Stacks
Definition:
A stack of size n ∈ N0 is a decomposition of n into positive integers
n = a1 + · · ·+ ar + c + bs + · · ·+ b1
such that
0 < a1 ≤ a2 ≤ · · · ≤ ar ≤ c < bs ≥ · · · ≥ b1 > 0.
s(n) := # stacks of size n
Example:
n = 4:4 3+1 1+3 2+2
2+1+1 1+2+1 1+1+2 1+1+1+1
Thus s(4) = 8.
Kathrin Bringmann Ramanujan and asymptotic formulas
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Generating function
Auluck:
S(q) :=∑n≥0
s(n)qn = 1 +∑n≥1
qn
(q)2n−1(1− qn)
(Non)-modularity:
S(q) = 1 +1
(q)2∞ϑ(q)
with the false theta function
ϑ(q) :=∑r≥1
(−1)r+1qr(r+1)
2
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Generating function
Auluck:
S(q) :=∑n≥0
s(n)qn = 1 +∑n≥1
qn
(q)2n−1(1− qn)
(Non)-modularity:
S(q) = 1 +1
(q)2∞ϑ(q)
with the false theta function
ϑ(q) :=∑r≥1
(−1)r+1qr(r+1)
2
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Asymptotic behavior
Wright:
s(n) ∼ 2−33−34 n−
54 e2π√
n3
Key ingredients:
I Modularity of the infinite product
I Asymptotic behavior as q → 1 of the false theta function
ϑ(q) ∼ 1
2as q → 1
I Tauberian Theorem
Kathrin Bringmann Ramanujan and asymptotic formulas
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Asymptotic behavior
Wright:
s(n) ∼ 2−33−34 n−
54 e2π√
n3
Key ingredients:
I Modularity of the infinite product
I Asymptotic behavior as q → 1 of the false theta function
ϑ(q) ∼ 1
2as q → 1
I Tauberian Theorem
Kathrin Bringmann Ramanujan and asymptotic formulas
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Asymptotic behavior
Wright:
s(n) ∼ 2−33−34 n−
54 e2π√
n3
Key ingredients:
I Modularity of the infinite product
I Asymptotic behavior as q → 1 of the false theta function
ϑ(q) ∼ 1
2as q → 1
I Tauberian Theorem
Kathrin Bringmann Ramanujan and asymptotic formulas
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Asymptotic behavior
Wright:
s(n) ∼ 2−33−34 n−
54 e2π√
n3
Key ingredients:
I Modularity of the infinite product
I Asymptotic behavior as q → 1 of the false theta function
ϑ(q) ∼ 1
2as q → 1
I Tauberian Theorem
Kathrin Bringmann Ramanujan and asymptotic formulas
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Asymptotic behavior
Wright:
s(n) ∼ 2−33−34 n−
54 e2π√
n3
Key ingredients:
I Modularity of the infinite product
I Asymptotic behavior as q → 1 of the false theta function
ϑ(q) ∼ 1
2as q → 1
I Tauberian Theorem
Kathrin Bringmann Ramanujan and asymptotic formulas
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Shifted stacks
Definition:
A shifted stack of size n ∈ N0 is a stack of size n with the extracondition that
aj ≥ aj+1 − 1 for 1 ≤ j ≤ r − 1,
bj ≤ bj+1 + 1 for 1 ≤ j ≤ s − 1.
ss(n) := #shifted stacks of size n
Example:
The shifted stacks of size 4 are
1 + 2 + 1 1 + 1 + 2 1 + 1 + 1 + 1
Thus ss(4) = 3.
Kathrin Bringmann Ramanujan and asymptotic formulas
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Shifted stacks
Definition:
A shifted stack of size n ∈ N0 is a stack of size n with the extracondition that
aj ≥ aj+1 − 1 for 1 ≤ j ≤ r − 1,
bj ≤ bj+1 + 1 for 1 ≤ j ≤ s − 1.
ss(n) := #shifted stacks of size n
Example:
The shifted stacks of size 4 are
1 + 2 + 1 1 + 1 + 2 1 + 1 + 1 + 1
Thus ss(4) = 3.
Kathrin Bringmann Ramanujan and asymptotic formulas
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Shifted stacks
Definition:
A shifted stack of size n ∈ N0 is a stack of size n with the extracondition that
aj ≥ aj+1 − 1 for 1 ≤ j ≤ r − 1,
bj ≤ bj+1 + 1 for 1 ≤ j ≤ s − 1.
ss(n) := #shifted stacks of size n
Example:
The shifted stacks of size 4 are
1 + 2 + 1 1 + 1 + 2 1 + 1 + 1 + 1
Thus ss(4) = 3.
Kathrin Bringmann Ramanujan and asymptotic formulas
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Shifted stacks
Definition:
A shifted stack of size n ∈ N0 is a stack of size n with the extracondition that
aj ≥ aj+1 − 1 for 1 ≤ j ≤ r − 1,
bj ≤ bj+1 + 1 for 1 ≤ j ≤ s − 1.
ss(n) := #shifted stacks of size n
Example:
The shifted stacks of size 4 are
1 + 2 + 1 1 + 1 + 2 1 + 1 + 1 + 1
Thus ss(4) = 3.
Kathrin Bringmann Ramanujan and asymptotic formulas
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Shifted stacks
Definition:
A shifted stack of size n ∈ N0 is a stack of size n with the extracondition that
aj ≥ aj+1 − 1 for 1 ≤ j ≤ r − 1,
bj ≤ bj+1 + 1 for 1 ≤ j ≤ s − 1.
ss(n) := #shifted stacks of size n
Example:
The shifted stacks of size 4 are
1 + 2 + 1
1 + 1 + 2 1 + 1 + 1 + 1
Thus ss(4) = 3.
Kathrin Bringmann Ramanujan and asymptotic formulas
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Shifted stacks
Definition:
A shifted stack of size n ∈ N0 is a stack of size n with the extracondition that
aj ≥ aj+1 − 1 for 1 ≤ j ≤ r − 1,
bj ≤ bj+1 + 1 for 1 ≤ j ≤ s − 1.
ss(n) := #shifted stacks of size n
Example:
The shifted stacks of size 4 are
1 + 2 + 1 1 + 1 + 2
1 + 1 + 1 + 1
Thus ss(4) = 3.
Kathrin Bringmann Ramanujan and asymptotic formulas
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Shifted stacks
Definition:
A shifted stack of size n ∈ N0 is a stack of size n with the extracondition that
aj ≥ aj+1 − 1 for 1 ≤ j ≤ r − 1,
bj ≤ bj+1 + 1 for 1 ≤ j ≤ s − 1.
ss(n) := #shifted stacks of size n
Example:
The shifted stacks of size 4 are
1 + 2 + 1 1 + 1 + 2 1 + 1 + 1 + 1
Thus ss(4) = 3.
Kathrin Bringmann Ramanujan and asymptotic formulas
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Shifted stacks
Definition:
A shifted stack of size n ∈ N0 is a stack of size n with the extracondition that
aj ≥ aj+1 − 1 for 1 ≤ j ≤ r − 1,
bj ≤ bj+1 + 1 for 1 ≤ j ≤ s − 1.
ss(n) := #shifted stacks of size n
Example:
The shifted stacks of size 4 are
1 + 2 + 1 1 + 1 + 2 1 + 1 + 1 + 1
Thus ss(4) = 3.
Kathrin Bringmann Ramanujan and asymptotic formulas
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Generating function
Auluck:
Ss(q) :=∑n≥0
ss(n)qn = 1 +∑n≥1
qn(n+1)
2
(q)2n−1(1− qn)
(log) asymptotic behavior (Wright)
log(ss(n)) ∼ 2π
√n
5(n→∞)
Theorem (B-Mahlburg)
We have
ss(n) ∼ φ−1
2√
2534 n
e2π√
n5 (n→∞)
with φ the Golden Ratio.
Kathrin Bringmann Ramanujan and asymptotic formulas
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Generating function
Auluck:
Ss(q) :=∑n≥0
ss(n)qn = 1 +∑n≥1
qn(n+1)
2
(q)2n−1(1− qn)
(log) asymptotic behavior (Wright)
log(ss(n)) ∼ 2π
√n
5(n→∞)
Theorem (B-Mahlburg)
We have
ss(n) ∼ φ−1
2√
2534 n
e2π√
n5 (n→∞)
with φ the Golden Ratio.
Kathrin Bringmann Ramanujan and asymptotic formulas
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Generating function
Auluck:
Ss(q) :=∑n≥0
ss(n)qn = 1 +∑n≥1
qn(n+1)
2
(q)2n−1(1− qn)
(log) asymptotic behavior (Wright)
log(ss(n)) ∼ 2π
√n
5(n→∞)
Theorem (B-Mahlburg)
We have
ss(n) ∼ φ−1
2√
2534 n
e2π√
n5 (n→∞)
with φ the Golden Ratio.
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 151: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/151.jpg)
Key idea of the proof
I Embedding into the modular world
Ss(q) = 1 + coeff[x0]∑
r≥0
x−rqr2−r
2
(q)r
∑m≥1
xmqm
(q)m−1
= 1 + coeff[x0] (
qx(−x−1
)∞ (xq)−1
∞
)I Use Jacobi triple product formula(
−x−1)∞ (−xq)∞ (q)∞ = −q−
18 x−
12ϑ
(u +
1
2;
iε
2π
)with the Jacobi theta function
ϑ(u; τ) :=∑
n∈ 12
+Z
eπin2τ+2πin(u+ 1
2 )
Kathrin Bringmann Ramanujan and asymptotic formulas
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Key idea of the proof
I Embedding into the modular world
Ss(q) = 1 + coeff[x0]∑
r≥0
x−rqr2−r
2
(q)r
∑m≥1
xmqm
(q)m−1
= 1 + coeff
[x0] (
qx(−x−1
)∞ (xq)−1
∞
)
I Use Jacobi triple product formula(−x−1
)∞ (−xq)∞ (q)∞ = −q−
18 x−
12ϑ
(u +
1
2;
iε
2π
)with the Jacobi theta function
ϑ(u; τ) :=∑
n∈ 12
+Z
eπin2τ+2πin(u+ 1
2 )
Kathrin Bringmann Ramanujan and asymptotic formulas
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Key idea of the proof
I Embedding into the modular world
Ss(q) = 1 + coeff[x0]∑
r≥0
x−rqr2−r
2
(q)r
∑m≥1
xmqm
(q)m−1
= 1 + coeff
[x0] (
qx(−x−1
)∞ (xq)−1
∞
)I Use Jacobi triple product formula(
−x−1)∞ (−xq)∞ (q)∞ = −q−
18 x−
12ϑ
(u +
1
2;
iε
2π
)with the Jacobi theta function
ϑ(u; τ) :=∑
n∈ 12
+Z
eπin2τ+2πin(u+ 1
2 )
Kathrin Bringmann Ramanujan and asymptotic formulas
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Proof continued . . .
I Also use quantum dilogarithm
Li2(x ; q) := − log(x)∞
I Modular inversion
ϑ
(u +
1
2;
iε
2π
)= i
√2π
εe−
2π2
ε (u+ 12 )
2
ϑ
(2π(u + 1
2
)iε
;2πi
ε
)I Laurent expansion of Li2
Li2(
e−Bεx ; e−ε)
=1
εLi2(x) +
(B − 1
2
)log(1− x) + O(ε).
I Saddle point method
I Tauberian Theorem
Kathrin Bringmann Ramanujan and asymptotic formulas
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Proof continued . . .
I Also use quantum dilogarithm
Li2(x ; q) := − log(x)∞
I Modular inversion
ϑ
(u +
1
2;
iε
2π
)= i
√2π
εe−
2π2
ε (u+ 12 )
2
ϑ
(2π(u + 1
2
)iε
;2πi
ε
)
I Laurent expansion of Li2
Li2(
e−Bεx ; e−ε)
=1
εLi2(x) +
(B − 1
2
)log(1− x) + O(ε).
I Saddle point method
I Tauberian Theorem
Kathrin Bringmann Ramanujan and asymptotic formulas
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Proof continued . . .
I Also use quantum dilogarithm
Li2(x ; q) := − log(x)∞
I Modular inversion
ϑ
(u +
1
2;
iε
2π
)= i
√2π
εe−
2π2
ε (u+ 12 )
2
ϑ
(2π(u + 1
2
)iε
;2πi
ε
)I Laurent expansion of Li2
Li2(
e−Bεx ; e−ε)
=1
εLi2(x) +
(B − 1
2
)log(1− x) + O(ε).
I Saddle point method
I Tauberian Theorem
Kathrin Bringmann Ramanujan and asymptotic formulas
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Proof continued . . .
I Also use quantum dilogarithm
Li2(x ; q) := − log(x)∞
I Modular inversion
ϑ
(u +
1
2;
iε
2π
)= i
√2π
εe−
2π2
ε (u+ 12 )
2
ϑ
(2π(u + 1
2
)iε
;2πi
ε
)I Laurent expansion of Li2
Li2(
e−Bεx ; e−ε)
=1
εLi2(x) +
(B − 1
2
)log(1− x) + O(ε).
I Saddle point method
I Tauberian Theorem
Kathrin Bringmann Ramanujan and asymptotic formulas
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Proof continued . . .
I Also use quantum dilogarithm
Li2(x ; q) := − log(x)∞
I Modular inversion
ϑ
(u +
1
2;
iε
2π
)= i
√2π
εe−
2π2
ε (u+ 12 )
2
ϑ
(2π(u + 1
2
)iε
;2πi
ε
)I Laurent expansion of Li2
Li2(
e−Bεx ; e−ε)
=1
εLi2(x) +
(B − 1
2
)log(1− x) + O(ε).
I Saddle point method
I Tauberian Theorem
Kathrin Bringmann Ramanujan and asymptotic formulas
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Positive crank and rank moments
Definition: For a partition λ define
rank(λ) := largest part of λ− number of parts of λ,
crank(λ) :=
{largest part of λ if o(λ) = 0,
µ(λ)− o(λ) if o(λ) > 0,
where
o(λ) := # of 1s in λ,
µ(λ) := #parts > o(λ).
Kathrin Bringmann Ramanujan and asymptotic formulas
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Positive crank and rank moments
Definition: For a partition λ define
rank(λ) := largest part of λ− number of parts of λ,
crank(λ) :=
{largest part of λ if o(λ) = 0,
µ(λ)− o(λ) if o(λ) > 0,
where
o(λ) := # of 1s in λ,
µ(λ) := #parts > o(λ).
Kathrin Bringmann Ramanujan and asymptotic formulas
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Positive crank and rank moments
Definition: For a partition λ define
rank(λ) := largest part of λ− number of parts of λ,
crank(λ) :=
{largest part of λ if o(λ) = 0,
µ(λ)− o(λ) if o(λ) > 0,
where
o(λ) := # of 1s in λ,
µ(λ) := #parts > o(λ).
Kathrin Bringmann Ramanujan and asymptotic formulas
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Rank and crank
Generating functions (basically)
M(m, n) := # of partitions of n, crank m
N(m, n) := # of partitions of n, rank m
Andrews–Garvan
C (w ; q) :=∑m∈Zn≥0
M(m, n)wmqn =1− w
(q)∞
∑n∈Z
(−1)nqn(n+1)
2
1− wqn,
Atkin–Swinnerton-Dyer
R(w ; q) :=∑m∈Zn≥0
N(m, n)wmqn =1− w
(q)∞
∑n∈Z
(−1)nqn(3n+1)
2
1− wqn.
Kathrin Bringmann Ramanujan and asymptotic formulas
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Rank and crank
Generating functions (basically)
M(m, n) := # of partitions of n, crank m
N(m, n) := # of partitions of n, rank m
Andrews–Garvan
C (w ; q) :=∑m∈Zn≥0
M(m, n)wmqn =1− w
(q)∞
∑n∈Z
(−1)nqn(n+1)
2
1− wqn,
Atkin–Swinnerton-Dyer
R(w ; q) :=∑m∈Zn≥0
N(m, n)wmqn =1− w
(q)∞
∑n∈Z
(−1)nqn(3n+1)
2
1− wqn.
Kathrin Bringmann Ramanujan and asymptotic formulas
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Rank and crank
Generating functions (basically)
M(m, n) := # of partitions of n, crank m
N(m, n) := # of partitions of n, rank m
Andrews–Garvan
C (w ; q) :=∑m∈Zn≥0
M(m, n)wmqn =1− w
(q)∞
∑n∈Z
(−1)nqn(n+1)
2
1− wqn,
Atkin–Swinnerton-Dyer
R(w ; q) :=∑m∈Zn≥0
N(m, n)wmqn =1− w
(q)∞
∑n∈Z
(−1)nqn(3n+1)
2
1− wqn.
Kathrin Bringmann Ramanujan and asymptotic formulas
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Moments
Crank and Rank moments (r ∈ N) (Atkin–Garvan)
Mr (n) :=∑m∈Z
mrM(m, n)
Nr (n) :=∑m∈Z
mrN(m, n)
Note: N2r+1(n) = M2r+1(n) = 0.
Theorem (B-Mahlburg, B-Mahlburg-Rhoades, Garvan)
(i) As n→∞M2k(n) ∼ N2k(n)
(ii) For all n > 0M2k(n) > N2k(n)
Important tool: modularity
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 166: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/166.jpg)
Moments
Crank and Rank moments (r ∈ N) (Atkin–Garvan)
Mr (n) :=∑m∈Z
mrM(m, n)
Nr (n) :=∑m∈Z
mrN(m, n)
Note: N2r+1(n) = M2r+1(n) = 0.
Theorem (B-Mahlburg, B-Mahlburg-Rhoades, Garvan)
(i) As n→∞M2k(n) ∼ N2k(n)
(ii) For all n > 0M2k(n) > N2k(n)
Important tool: modularity
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 167: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/167.jpg)
Moments
Crank and Rank moments (r ∈ N) (Atkin–Garvan)
Mr (n) :=∑m∈Z
mrM(m, n)
Nr (n) :=∑m∈Z
mrN(m, n)
Note: N2r+1(n) = M2r+1(n) = 0.
Theorem (B-Mahlburg, B-Mahlburg-Rhoades, Garvan)
(i) As n→∞M2k(n) ∼ N2k(n)
(ii) For all n > 0M2k(n) > N2k(n)
Important tool: modularity
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 168: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/168.jpg)
Moments
Crank and Rank moments (r ∈ N) (Atkin–Garvan)
Mr (n) :=∑m∈Z
mrM(m, n)
Nr (n) :=∑m∈Z
mrN(m, n)
Note: N2r+1(n) = M2r+1(n) = 0.
Theorem (B-Mahlburg, B-Mahlburg-Rhoades, Garvan)
(i) As n→∞M2k(n) ∼ N2k(n)
(ii) For all n > 0M2k(n) > N2k(n)
Important tool: modularity
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 169: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/169.jpg)
Moments
Crank and Rank moments (r ∈ N) (Atkin–Garvan)
Mr (n) :=∑m∈Z
mrM(m, n)
Nr (n) :=∑m∈Z
mrN(m, n)
Note: N2r+1(n) = M2r+1(n) = 0.
Theorem (B-Mahlburg, B-Mahlburg-Rhoades, Garvan)
(i) As n→∞M2k(n) ∼ N2k(n)
(ii) For all n > 0M2k(n) > N2k(n)
Important tool: modularity
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 170: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/170.jpg)
Positive moments
Positive crank and rank moments (r ∈ N) (Andrews–Chan–Kim)
M+r (n) :=
∑m∈N
mrM(m, n)
N+r (n) :=
∑m∈N
mrN(m, n)
Note: M+2k(n) = 1
2 M2k(n)
N+2k(n) = 1
2 N2k(n)
Theorem (B-Mahlburg, Andrews-Chan-Kim)
(i) As n→∞M+
r (n) ∼ N+r (n)
(ii) For all n > 0M+
r (n) > N+r (n)
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 171: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/171.jpg)
Positive moments
Positive crank and rank moments (r ∈ N) (Andrews–Chan–Kim)
M+r (n) :=
∑m∈N
mrM(m, n)
N+r (n) :=
∑m∈N
mrN(m, n)
Note: M+2k(n) = 1
2 M2k(n)
N+2k(n) = 1
2 N2k(n)
Theorem (B-Mahlburg, Andrews-Chan-Kim)
(i) As n→∞M+
r (n) ∼ N+r (n)
(ii) For all n > 0M+
r (n) > N+r (n)
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 172: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/172.jpg)
Positive moments
Positive crank and rank moments (r ∈ N) (Andrews–Chan–Kim)
M+r (n) :=
∑m∈N
mrM(m, n)
N+r (n) :=
∑m∈N
mrN(m, n)
Note: M+2k(n) = 1
2 M2k(n)
N+2k(n) = 1
2 N2k(n)
Theorem (B-Mahlburg, Andrews-Chan-Kim)
(i) As n→∞M+
r (n) ∼ N+r (n)
(ii) For all n > 0M+
r (n) > N+r (n)
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 173: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/173.jpg)
Positive moments
Positive crank and rank moments (r ∈ N) (Andrews–Chan–Kim)
M+r (n) :=
∑m∈N
mrM(m, n)
N+r (n) :=
∑m∈N
mrN(m, n)
Note: M+2k(n) = 1
2 M2k(n)
N+2k(n) = 1
2 N2k(n)
Theorem (B-Mahlburg, Andrews-Chan-Kim)
(i) As n→∞M+
r (n) ∼ N+r (n)
(ii) For all n > 0M+
r (n) > N+r (n)
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 174: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/174.jpg)
Idea of proof
Difficulty: non-modularity
I Relate generating functions to “false Lerch sums” (` ∈ {1, 3})
1
(q)∞
∑n≥1
(−1)n+1q`n2
2+ rn
2
(1− qn)r
I Understand the asymptotic behavior near q = 1
I Use Circle Method
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 175: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/175.jpg)
Idea of proof
Difficulty: non-modularity
I Relate generating functions to “false Lerch sums” (` ∈ {1, 3})
1
(q)∞
∑n≥1
(−1)n+1q`n2
2+ rn
2
(1− qn)r
I Understand the asymptotic behavior near q = 1
I Use Circle Method
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 176: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/176.jpg)
Idea of proof
Difficulty: non-modularity
I Relate generating functions to “false Lerch sums” (` ∈ {1, 3})
1
(q)∞
∑n≥1
(−1)n+1q`n2
2+ rn
2
(1− qn)r
I Understand the asymptotic behavior near q = 1
I Use Circle Method
Kathrin Bringmann Ramanujan and asymptotic formulas
![Page 177: Ramanujan and asymptotic formulas - IRIFsteiner/jifp/bringmann.pdf · Kathrin Bringmann Ramanujan and asymptotic formulas. Outline 1.Modular forms 2.Mock modular forms 3.Mixed mock](https://reader033.fdocuments.in/reader033/viewer/2022050601/5fa8f09c4aa463368d7f6083/html5/thumbnails/177.jpg)
Idea of proof
Difficulty: non-modularity
I Relate generating functions to “false Lerch sums” (` ∈ {1, 3})
1
(q)∞
∑n≥1
(−1)n+1q`n2
2+ rn
2
(1− qn)r
I Understand the asymptotic behavior near q = 1
I Use Circle Method
Kathrin Bringmann Ramanujan and asymptotic formulas