Synopsis of a Plenary talk at the The Legacy of Ramanujan conference in honor of George Andrews' and Bruce Berndt's 85th birthdays, Friday June 7, 2024, 1:30-2:20 pm. By Doron Zeilberger
Abstract: The three heroes, in chronological order of birth, are Srinivasa Ramanujan (born Dec. 22, 1887), Bruce Berndt (born March 13, 1938, (sic!)), and George Andrews (born Dec. 4, 1938). The three masterpieces were written when they were 24, 28, and 22 years-old, respectively
This great conference celebrates these three heroes, namely Srinivasa R., George A., and Bruce B., and my talk would focus on their work when they were (really) young men. According to Ramanujan's mentor G.H. Hardy, "mathematics is a young man's game", and while this is not the only stupid and false thing that he said (see below for an even stupider statement), witness that the two birthday boys, both over 85 years old, are as active and prolific as ever, and hence offer clear refutation to Professor Hardy. Nevertheless, it is still instructive, and fun, to go back and read their very early papers, and realize that they already showed promise way back then, that was later on fulfilled with a vengeance.
Speaking of Hardy, he famously wrote a manifesto entitled a mathematician's aplogy, (that I will go back to later). Speaking of apology, I owe you two of them.
"If I came to your talk, I expect you to come to my talk."
In my case, I believe that it was an excused absence, since I really wanted to be present at my hero Don Knuth's banquet talk at the Stanley@80 conference, that disappointingly had a non-empty intersection with this conference. I promise to carefully read the slides of yesterday's talks once they are posted.
Now let's start the actual talk, since now I know that Bruce is the youngest of these three heroes, let's start with him. Bruce wrote a brilliant PhD thesis, at the University of Wisconsin, under Rod Smart, about Hecke functional equations, that was written up in this masterpiece. This is analytic number theory at its best. A few years later he wrote yet another paper on functional equations that had the following delightful motto:
"One bright Sunday morning I went to church,
And there I met a man named Lerch.
We both did sing in jubilation,
For he did show me a new equation."
[during the talk, Bruce narrated that a few years after the paper was published, he had a Czech visitor, Bretislav Novak, who told him that he found a "serious mistake". Bruce, obviously terrified, asked that visitor to tell him what was wrong with the paper. Novak replied that in Czech (Mathias Lerch's native language), the "ch" is silent, so the name is pronounced "ler", and hence does not ryhme with "Church". After the talk Bruce told me that the then editor of the journal (Proc. of the AMS),Joseph Rotman, who also happened to be his colleague at U. if I., met him in the mailroom and told him: "I hope that you are not going to publish this doggerel", but Bruce persisted, and won. When I was a postdoc at UIUC, in 1979 (when I met Bruce for the first time), I remember the 1979 math department Christmas party, when Bruce read a parody on "twas the night before Christmas". Bruce told me that he continues with this tradition to this day, and his recent creation is a poem about Donald Trump, that he kindly promised to email me, and I am looking forward to it.]
As Bruce will tell you in his talk on Sunday, his "labor of love", for which he dedicated most of his later professional life, started when he was a member of the Institute for Advanced Study, when he met Robert Rankin and he challenged him to prove some statements in Ramanujan's (the original, not the lost one) notebook, that he did rather quickly, but then found others that he could not. This lead, in 1985, to the First volume of Ramanujan's Notebooks, [I am proud to have (very modestly) contributed to this first volume, please do "^F Zeilberger" to see what I did] that was followed by four more volumes, and together with the other birthday boy, George Andrews, five volumes of Ramanujan's lost notebooks.
Let's now go to my second hero, George Andrews, and try to see and understand the masterpiece, that was part of his master's thesis at what was then called "Oregon State College", that is now a major research university, but then must have been a predominantly teaching institution. As you can see (please at least browse it, but I strongly recommend that you read it carefully) this aleady had seeds of his later seminal work. I particularly like this follow-up Monthly note, where he used the original result to prove elementary upper bounds for d(n), the number of divisors of n. Namley d(n) ≤ n1/3log(n). George, was at the time a PhD student at Penn, under the great Hans Rademacher.
As we all know, both Bruce and George went on to have produced many more great results in mathematics, and they at 85, are living refutations to Hardy's "young man's game", but now we turn to refuting another erroneous statement by the great G.H. Hardy, whom I love to hate, but still forgive him, since he was so kind to Ramanujan.
Hardy was a "pure math snob" and was so happy that "number theory is useless", hence his insincere apology for not doing anything "productive". He also thought that applied math was "ugly", and in some sense, "trivial".
Fast forward to the space age, when humankind needed pictures from Mars. Since communication is noisy, we need error correcting codes, and this beautiful part of applied math, uses finite fields, and number theory, in a non-trivial way (of course now we also have RSA as another example). This was already pointed out in Normal Levinson's (another hero of mine) defense of applied math, but yet another example, not mentioned by Levinson, is the application to so-called BCH codes (named after Bose, Ray-Chaudhuri, and Hocquenghem). See chapter 11 of this delightful Coding Theory book, by Raymond Hill.
Anyway, in order to decode BCH codes, what was needed was this two-page Masterpiece by 24-year-old Srinivasa Ramanujan
Please read it carefully. It is really neat! He converts a system of 2n non-linear equations into 2n linear equations, and then uses partial fractions that we all teach in calc2, to recover the solutions to the original system. Here is Maple code for Ramanujan's method
To do Ramanujan's example type
lprint(simplify(aSr([2,3,16,31,103,235,674,1669,4526,11595])));
getting
[-3/5, 1/5*(111-47*5^(1/2))/(5^(1/2)-3)^2, 1/5*(111+47*5^(1/2))/(3+5^(1/2))^2,
(-29+11*5^(1/2))/(5^(1/2)-1)/(-5+5^(1/2)), (-29-11*5^(1/2))/(5^(1/2)+1)/(5+5^(1
/2))], [-1, -2/(5^(1/2)-3), 2/(3+5^(1/2)), -2/(5^(1/2)-1), 2/(5^(1/2)+1)]]
To get it in decimals, type:
lprint(evalf(aSr([2,3,16,31,103,235,674,1669,4526,11595])));
getting
[[-.6000000000, 2.023606792, 1.576393202, 1.288854381, -2.288854383], [-1., 2.6\
18033984, .3819660114, -1.618033991, .6180339890]]
Of course, when you apply this method for decoding BCH codes, the operations are all over a finite field, and are even easier.
Attendance Quiz: Since this "lecture" was really meant to be a "class", where I really wanted you to learn and understand Ramanujan's method, I gave, like in all my classes, an attendance quiz, that does not count towards the grade, but is to test your understanding. I offered to give prizes to the three best solutions, a copy of Ken Ono's fascinating memoir. The winners are listed below. I will email them and ask for their snail-mail addresses.
Use Ramanujan's method (no credit for inspection or other methods!) to solve the following system of four equations and four unknowns.
x1+ x2=2
x1y1+ x2y2=0
x1y12+ x2y22=2
x1y13+ x2y23=0
And the winners are:
Personal Journal of Shalosh B. Ekhad and Doron Zeilberger