"The Riemann Hypothesis" by J.E. Littlewood I believe this to be false. There is no evidence whatever for it (unless one counts that it is always nice when any function has only real roots). One should not believe things for which there is no evidence. In the spirit of this anthology I should also record my feeling that there is no imaginable reason why it should be true. Titchmarsh [1] devised a method, of considerable theoretical interest, for calculating the zeros. The method reveals that for a zero to be off the critical line a remarkable number of 'coincidences' have to happen. I have discussed the matter with several people who know the problem in relation to electronic calculation; they are all agreed that the chance of finding a zero off the line in a lifetime's calculation is millions to one against. It looks then as if we may never know. It is true that the existence of an infinity of L-functions raising the same problems creates a remarkable situation. Nonetheless life would be more comfortable if one could believe firmly that the hypothesis is false. Partly in response to the above, one of the anthology's editors, I.J. Good, in collaboration with R.F. Churchhouse, published the brief article "The Riemann hypothesis and pseudorandom features of the Möbius sequence" [2], the stated aim of which is "to suggest a "reason" for believing Riemann's hypothesis". The authors make use of probabilistic reasoning and a Chilton Atlas computer (presumably one of the fastest available in 1968). Abstract. A study of the cumulative sums of the Möbius function on the Atlas Computer of the Science Research Council has revealed certain statistical properties which lead the authors to make a number of conjectures. One of these is that any conjecture of the Mertens type, viz. where k is any positive constant, is false, and indeed the authors conjecture that [1] E.C. Titchmarsh, The Riemann Zeta-Function, Oxford, 1951. [2] I.J. Good and R.F. Churchhouse, "The Riemann hypothesis and pseudorandom features of the Möbius sequence", Mathematics of Computation 22 (1968) 857-864. Note from Andrew Odlyzko (February 1, 2000): "Concerning Littlewood, I am working on a monograph on computations of zeros of the zeta function , which will feature an extended discussion of various conjectures and the numerical evidence for them. Perhaps when I am done I might be willing to write a commentary on Littlewood's opinions. Personally I am a bit of an agnostic about the R.H., but would definitely not agree with Littlewood to the fullest. (BTW, I am familiar with the Good–Churchhouse argument, and have pointed out in one of my papers that while it is appealing, it definitely fails some important tests.)" number theory and physics archive prime numbers: FAQ and resources mystery new search home |