The Riemann hypothesis and pseudorandom features of the Möbius sequence
Authors:
I. J. Good and R. F. Churchhouse
Journal:
Math. Comp. 22 (1968), 857861
MSC:
Primary 10.41
MathSciNet review:
0240062
Fulltext 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. where is any positive constant, is false, and indeed the authors conjecture that .
 [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
(13,363e)
 [3]
C.
B. Haselgrove, A disproof of a conjecture of Pólya,
Mathematika 5 (1958), 141–145. MR 0104638
(21 #3391)
 [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
(91f:11064)
 [5]
J. E. Littlewood, "Sur la distribution des nombres premiers," C. R. Acad. Sci. Paris, v. 158, 1914, pp. 18691872.
 [6]
J. E. Littlewood, "The Riemann hypothesis" in The Scientist Speculates, edited by Good, Mayne & Maynard Smith, London and New York, 1962, pp. 390391.
 [7]
Gerhard
Neubauer, Eine empirische Untersuchung zur Mertensschen
Funktion, Numer. Math. 5 (1963), 1–13 (German).
MR
0155787 (27 #5721)
 [8]
J. B. Rosser & L. Schoenfeld, "The first two million zeros of the Riemann zetafunction are on the critical line," Abstracts for the Conference of Mathematicians, Moscow, 1966, 8. (Unpublished.)
 [9]
E.
C. Titchmarsh, The Theory of the Riemann ZetaFunction,
Oxford, at the Clarendon Press, 1951. MR 0046485
(13,741c)
 [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. 756762. MR 13, 363. MR 0044061 (13:363e)
 [3]
 C. B. Haselgrove, "A disproof of a conjecture of Pólya," Mathematika, v. 5, 1958, pp. 141145. 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. 18691872.
 [6]
 J. E. Littlewood, "The Riemann hypothesis" in The Scientist Speculates, edited by Good, Mayne & Maynard Smith, London and New York, 1962, pp. 390391.
 [7]
 G. Neubauer, "Eine empirische Untersuchung zur Mertensschen Funktion," Numer. Math., v. 5, 1963, pp. 113. MR 27 #5721. MR 0155787 (27:5721)
 [8]
 J. B. Rosser & L. Schoenfeld, "The first two million zeros of the Riemann zetafunction are on the critical line," Abstracts for the Conference of Mathematicians, Moscow, 1966, 8. (Unpublished.)
 [9]
 E. C. Titchmarsh, The Theory of the Riemann ZetaFunction, 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
DOI:
http://dx.doi.org/10.1090/S00255718196802400628
PII:
S 00255718(1968)02400628
Article copyright:
© Copyright 1968 American Mathematical Society
