Date: Fri, 03 Sep 1999 16:39:32 -0700 To: firstname.lastname@example.org (Antreas P. Hatzipolakis) From: Tom Apostol <email@example.com> Subject: Re: S. Mac Lane sang a song about Riemann
At 10:41 PM 9/3/99 +0300, you wrote:
>On Sun, 22 Aug, I quoted John Baez: > > in Portugal on Category Theory >>The conference was a big deal this year, because it celebrated the >>90th birthday of Saunders Mac Lane, who with Samuel Eilenberg invented >>category theory in 1945. Mac Lane was there and in fine fettle. He >>gave a nice talk about working with Eilenberg, and after the banquet >>in his honor, he even sang a song about Riemann while wrapped in a >>black cloak! > >On Thu, 02 Sep 1999, Scott Williams wrote: > >| I just read your Forum letter. My buddies Bill Lawvere and Steve >| Schanuel also gave me a Portugal report. > >But which was the song ? The following one by Tom Apostol<opoulos>, perhaps? > > zeta(s) > > The Zeta Function Song > > (Sung to the tune of "Sweet Betsy from Pike") > > Where are the zeros of zeta of s? > G. F. B. Riemann has made a good guess, > They're all on the critical line, said he, > And their density's one over 2 pi log t. > > This statement of Riemann's has been like a trigger, > And many good men, with vim and with vigor, > Have attempted to find, with mathematical rigor, > What happens to zeta as mod t gets bigger. > > The names of Landau and Bohr and Cramer, > And Hardy and Littlewood and Titchmarsh are there, > In spite of their efforts and skill and finesse, > In locating the zeros no one's had success. > > In 1914 G. H. Hardy did find, > An infinite number that lay on the line, > His theorem, however, won't rule out the case, > That there might be a zero at some other place. > > Let P be the function pi minus li, > The order of P is not known for x high, > If square root of x times log x we could show, > Then Riemann's conjecture would surely be so. > > Related to this is another enigma, > Concerning the Lindelhof function mu(sigma) > Which measures the growth in the critical strip, > And on the number of zeros it gives us a grip. > > But nobody knows how this function behaves, > Convexity tells us it can have no waves, > Lindelhof said that the shape of its graph, > Is constant when sigma is more than one half. > > Oh, where are the zeros of zeta of s? > We must know exactly, we cannot just guess, > In order to strengthen the prime number theorem, > The path of integration must not get too near'em. > >
For your information, the foregoing was written (and performed) by me on the occasion of the Number Theory Conference held at Caltech in June 1955.
My verses stimulated some unknown bard to post the following lines on the bulletin board at Cambridge University in 1973:
What Tom Apostol Didn't Know
Andre Weil has bettered old Riemann's fine guess, By using a fancier zeta of s, He proves that the zeros are where they should be, Provided the characteristic is p.
There's a good moral to draw from this long tale of woe That every young genius among you should know: If you tackle a problem and seem to get stuck, Just take it mod p and you'll have better luck.
In a telephone conversation with Tom Apostol in 1990, Saunders MacLane claimed to be the author of the two verses posted anonymously in Cambridge. MacLane has since added the following new verses:
What fraction of zeros on the line will be found When mod t is kept below some given bound? Does the fraction, whatever, stay bounded below As the bound on mod t is permitted to grow?
The efforts of Selberg did finally banish All fears that the fraction might possibly vanish. It stays bounded below, which is just as it should, But the bound he determined was not very good.
Norm Levinson managed to show, better yet, At two-to-one odds it would be a good bet, If over a zero you happen to trip It would lie on the line and not just in the strip.
Levinson tried in a classical way, Weil brought modular means into play, Atiyah then left and Paul Cohen quit, So now there's no proof at all that will fit.
But now we must study this matter anew, Serre points out manifold things it makes true, A medal might be the reward in this quest, For Riemann's conjecture is surely the best.
MacLane may have written more verses that I don't know about.
PS: The story of the long Greek (and others) surnames truncation in the States (Apostolopoulos --> Apostol):
In the following article: Kageleiry, Jamie: What Hardly Anyone Knows About Ellis Island. The Old Farmer's Almanac 200(1992) pp. 76 - 79
we read how the immigrants' names got transmogrified by officials at Ellis Island (during the years 1892-1954)
Some examples of long surnames: Andrjuljawierjus - Grzyszczyszn - Koutsoghianopoulos - Zemiskicivicz this is Greek