Showing papers in "Proceedings of The London Mathematical Society in 1953"

Some Calculations of the Riemann Zeta-Function

Abstract: Introduction IN June 1950 the Manchester University Mark 1 Electronic Computer was used to do some calculations concerned with the distribution of the zeros of the Riemann zeta-function. It was intended in fact to determine whether there are any zeros not on the critical line in certain particular intervals. The calculations had been planned some time in advance, but had in fact to be carried out in great haste. If it had not been for the fact that the computer remained in serviceable condition for an unusually long period from 3 p.m. one afternoon to 8 a.m. the following morning it is probable that the calculations would never have been done at all. As it was, the interval 2TT.63 < t < 2?r.64 was investigated during that period, and very little more was accomplished. The calculations were done in an optimistic hope that a zero would be found off the critical line, and the calculations were directed more towards finding such zeros than proving that none existed. The procedure was such that if it had been accurately followed, and if the machine made no errors in the period, then one could be sure that there were no zeros off the critical line in the interval in question. In practice only a few of the results were checked by repeating the calculation, so that the machine might well have made an error. If more time had been available it was intended to do some more calculations in an altogether different spirit. There is no reason in principle why computation should not be carried through with the rigour usual in mathematical analysis. If definite rules are laid down as to how the computation is to be done one can predict bounds for the errors throughout. When the computations are done by hand there are serious practical difficulties about this. The computer will probably have his own ideas as to how certain steps should be done. When certain steps may be omitted without serious loss of accuracy he will wish to do so. Furthermore he will probably not see the point of the 'rigorous' computation and will probably say 'If you want more certainty about the accuracy why not just take more figures?' an argument difficult to counter. However, if the calculations are being done by an automatic computer one can feel sure that this kind of indiscipline

Local Topological Invariants

Abstract: was to consider relationships between the local groups occurring in the Vietoris, singular and homotopy theories. The referee suggested that the local "C" and "D" groups occurring in the theory and defined in [5] and [6] (hereafter referred to as LTI and CTM respectively), were not "functorial" in the sense that the isomorphisms connected with them were merely "abstract," not induced by maps of one space into another and so not natural. He outlined a new approach using inverse and direct systems of groups, and in many cases the limits of these were isomorphic to the corresponding "C" and "D" groups; but in some cases the limits gave the "wrong" results. To overcome this, he suggested the idea of a stable system, where to postulate stability is to postulate something rather stronger than, but often equivalent to, existence of the "C" and "D" groups. (In locally Euclidean spaces and the generalized manifolds of Wilder [15], stability occurs at each point in each dimension.) We have therefore re-cast the whole of our previous theory in terms of these new concepts, thereby obtaining a more harmonious theory than before; and many of the results of the earlier draft together with analogues of results in LTI and CTM are here obtained. The plan of the paper is as follows. There are four sections: in §1 we prove all the basic results we later need on inverse and direct systems of groups, concerning their "stability" under mappings of various sorts. §11 is devoted to a discussion of certain relationships berween Singular and Vietoris homology. In §111, we derive certain results concerning homotopy, which are applied in §IV with the earlier ones to prove theorems concerning the local groups there. Corollaries of theorems in II and III give useful global results of the form:—if XCY, then under certain conditions and with different values of the functor G, the image of the injection G(X)—>G(Y) is finitely generated (see 2.33,3.14, 3.15). §IV is concerned essentially with three matters: first the proof that the Wilder manifolds, as mentioned above, have the stability property; second, implications between the various types of local connectivity, with some pathology; and third, proofs that for Singular and Vietoris homology, all the local groups we define (using stability) give the same end-product, i.e. the same class of manifolds,—with a similar but more restricted result for homotopy. Moreover, a "local" theorem of Hurewicz type is proved in 4.35.