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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?)

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Article copyright: © Copyright 1968 American Mathematical Society

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