Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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., v. 47, 1951, pp. 756-762. MR 13, 363. MR 0044061 (13:363e)
  • [3] C. B. Haselgrove, "A disproof of a conjecture of Pólya," Mathematika, v. 5, 1958, pp. 141-145. MR 21 #3391. MR 0104638 (21:3391)
  • [4] A. E. Ingham, The Distribution of Prime Numbers, Cambridge Univ. Press, New York, 1932. MR 1074573 (91f:11064)
  • [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] G. Neubauer, "Eine empirische Untersuchung zur Mertensschen Funktion," Numer. Math., v. 5, 1963, pp. 1-13. MR 27 #5721. MR 0155787 (27:5721)
  • [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 (Unpublished.)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, Clarendon Press, Oxford, 1951. MR 13, 741. MR 0046485 (13:741c)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 10.41

Retrieve articles in all journals with MSC: 10.41

Additional Information

Article copyright: © Copyright 1968 American Mathematical Society

American Mathematical Society