Littlewood's Tauberian theorem and Dyson's ferromagnet paper

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FREEMAN DYSON

26 - Littlewood's Tauberian theorem and Dyson's ferromagnet paper

He wrote this marvellous paper about the Tauberian theorem, the (1/n) theorem, which tells that a series that is summable in the Abel sense, and has terms bounded by a constant times (1/n) is also convergent. So that was the big Tauberian theorem which Littlewood proved, and this way of proving it was again a tour de force done by invoking a completely unmotivated new parameter which turned out to be the key to the proof. And then in my existence as a physicist one of the things I did was to prove the existence of a ferromagnet in one dimension, which was something that hadn't been done before, and I constructed this ferromagnet - this was a very 'Besicovitchy', a sort of Besicovitch construction of the ferromagnet as a hierarchical array of spins which has a ferromagnetic transition from order to disorder in spite of being one dimensional. And then, in order to prove the long range order - what I proved was that the long range order is discontinuous at the transition, that it goes discontinuously from having a spontaneous magnetisation to being disordered, so it means the ordered phase has a magnetisation which remains finite away from zero right up to the transition point. And to prove that, I used Littlewood's trick of bringing a unmotivated new parameter - so it was a proof; although the construction was Besicovitch, the proof was Littlewood. Anyway, so I dedicated this paper to him and it was written just, I think, 60 years after Littlewood's proof had been published and so I dedicated it to the 60th anniversary of Littlewood's proof. And I got a very nice letter from Littlewood in response, saying that as far as he knew that was the only time that a mathematician had actually celebrated the 60th anniversary of a proof. But he also said very complimentary things; that it was the most complimentary statement made to him by anyone, am I right? Yes, something like that; I doubt whether that was true, but anyway it was a very gracious acknowledgment. No, I was delighted that he was still around when this happened. Whereas with Hardy - of course Hardy died much younger, I was never able to thank Hardy for the Jordan's Cours d'analyse which I'm sure he actually put in the library, but he died before we could thank him for it.

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16 Freeman Dyson
Physicist

GHH Hardy and JE Littlewood's lectures

Keywords: JE Littlewoods; GH Hardy

Norman Routledge
Teacher

I wish I had met GH Hardy

Keywords: GH Hardy

Michael Atiyah
Mathematician

First year at Cambridge

Keywords: Abram Besicovitch

Francis Crick
Biologist

What makes a good collaboration?

Keywords: GH Hardy

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21. Political feelings in the '30s: Communism and pacifism

22. Being on the outside

23. Pure mathematics at Cambridge: the influence of Besicovitch

24. Relationship with Besicovitch

25. GHH Hardy and JE Littlewood's lectures

26. Littlewood's Tauberian theorem and Dyson's ferromagnet paper

27. Other tutors at Cambridge: Dirac, Jeffreys, Eddington

28. Oscar Hahn

29. The Hahn Family - Judaism

30. The walk from Cambridge to London

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Freeman Dyson
Physicist

GHH Hardy and JE Littlewood's lectures

Norman Routledge
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I wish I had met GH Hardy

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GHH Hardy and JE Littlewood's lectures

I wish I had met GH Hardy

First year at Cambridge

What makes a good collaboration?

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