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Tuesday, February 14, 2006

Making reference

Further nuances appear today in the statistical analysis of the Riemann zeta zeros. The evidence looks strong for the conjecture that in the limit as one climbs the line Re(z) = 1/2, the distribution of normalised spacings between zeros tends to that of the eigenvalues of large random matrices from the Unitary Ensemble. Bogomolny and colleagues show how as we pass up Re(z) = 1/2 away from the real axis, the deviation from the asymptote resembles that of the distribution of eigenvalues from finite dimensional matrices. They calculate the effective dimension of these matrices in relation to the height above the real axis, and verify their claims by studying batches of zeros, including some around the 1.3 x 1022 th. Elsewhere, the first ten trillion consecutive zeros have been checked.

There's plenty to interest a philosopher about this work - e.g., Would it be right to say that with the advent of computers mathematics has become 'empirical'? - but the issue I raise in my 'Mathematical Kinds' paper (p. 22) points elsewhere. There's a huge amount of work going on aimed at explaining why the Riemann zeros are distributed as they are. Berry and Keating observe:

… three areas of mathematics and physics, usually regarded as separate, are intimately connected. The analogy is tentative and tantalizing, but nevertheless fruitful. The three areas are eigenvalue asymptotics in wave (and particular quantum) physics, dynamic chaos, and prime number theory. At the heart of the analogy is a speculation concerning the zeros of the Riemann zeta function (an infinite sequence of numbers encoding the primes): the Riemann zeros are related to the eigenvalues (vibration frequencies, or quantum energies) of some wave system, underlying which is a dynamical system whose rays of trajectories are chaotic.

Identification of this dynamical system would lead directly to a proof of the celebrated Riemann hypothesis. We do not know what the system is, but we do know many of its properties, and this knowledge has brought insights in both directions...

Berry B. and Keating J. (1999) ‘The Riemann Zeros and Eigenvalue Asymptotics’ SIAM Review, 41, No.2 , 236-266.

What are we to make of the sense they give that there is a dynamical system, a mathematical entity, many of whose properties are known, but which is not yet completely identified? Similar themes have debated in the philosophy of science - e.g., when should it be said that a scientist made proper reference to the electron? Here, if Berry and Keating's system is later identified, what will decide whether they had already made reference to it? Another excellent case study would look at the question of when it could be said that sufficient theoretical resources had been provided to make reference to the monster group. Merely conjecturing that there is a largest sporadic finite simple group would surely not suffice. On the other hand, now, even if we don't yet completely understand it, we clearly have it pegged.


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