404 | 36 INTRODUCING TURING’S mATHEmATICS
Conclusion
If the actual statement of the Riemann hypothesis seems an anticlimax after the big build-up, its
consequences are substantial. Recalling Riemann’s discovery of the role that the zeta function’s
zeros play in the prime-counting function p(x), and his exact formula (involving the zeta func-
tion) for the number of primes up to x, we note that any divergence of these zeros from the line
x =^12 would crucially affect Riemann’s exact formula, since our understanding about how the
prime numbers behave is tied up in this formula. If Riemann’s hypothesis fails, then the prime
number theorem would still be true but would lose its command of the primes—instead of Don
Zagier’s ‘military precision’, the primes would be found to be in full mutiny!
We conclude with an unexpected development. In 1972 the American number theorist
Hugh Montgomery was visiting the tea room at Princeton’s Institute for Advanced Study and
found himself sitting opposite the celebrated physicist Freeman Dyson. Montgomery had been
exploring the spacings between the zeros on the critical line, and Dyson said: ‘But those are
just the spacings between the energy levels of a quantum chaotic system’. If this analogy indeed
holds, as many think possible, then the Riemann hypothesis may well have consequences in
quantum physics. Conversely, using their knowledge of these energy levels, quantum physicists
rather than mathematicians may be the ones to prove the Riemann hypothesis. Turing would
surely have been delighted with such an observation!