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The Riemann hypothesis and pseudorandom features of the Möbius sequence

Authors: I. J. Good and R. F. Churchhouse
Journal: Math. Comp. 22 (1968), 857-861
MSC: Primary 10.41
MathSciNet review: 0240062
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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.

$\displaystyle \vert M(N)\vert = \left\vert {\sum\limits_{n = 1}^N {\mu (n)} } \right\vert < k\left( {\surd N} \right)$

where $ k$ is any positive constant, is false, and indeed the authors conjecture that

$\displaystyle {\text{Lim}}\sup \left\{ {M(x){{(x\log \log x)}^{ - 1/2}}} \right... {{\surd \left( {12} \right)} \pi }} \right. \kern-\nulldelimiterspace} \pi }$


References [Enhancements On Off] (What's this?)

  • [1] W. K. Feller, An Introduction to Probability Theory and its Applications, Vol. 1, Wiley, New York, 1950. MR 12, 424.
  • [2] I. J. Good, Random motion on a finite Abelian group, Proc. Cambridge Philos. Soc. 47 (1951), 756–762. MR 0044061
  • [3] C. B. Haselgrove, A disproof of a conjecture of Pólya, Mathematika 5 (1958), 141–145. MR 0104638,
  • [4] A. E. Ingham, The distribution of prime numbers, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1990. Reprint of the 1932 original; With a foreword by R. C. Vaughan. MR 1074573
  • [5] J. E. Littlewood, "Sur la distribution des nombres premiers," C. R. Acad. Sci. Paris, v. 158, 1914, pp. 1869-1872.
  • [6] J. E. Littlewood, "The Riemann hypothesis" in The Scientist Speculates, edited by Good, Mayne & Maynard Smith, London and New York, 1962, pp. 390-391.
  • [7] Gerhard Neubauer, Eine empirische Untersuchung zur Mertensschen Funktion, Numer. Math. 5 (1963), 1–13 (German). MR 0155787,
  • [8] J. B. Rosser & L. Schoenfeld, "The first two million zeros of the Riemann zeta-function are on the critical line," Abstracts for the Conference of Mathematicians, Moscow, 1966, 8. (Unpublished.)
  • [9] E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, Oxford, at the Clarendon Press, 1951. MR 0046485

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Article copyright: © Copyright 1968 American Mathematical Society

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