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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Sign changes in sums of the Liouville function


Authors: Peter Borwein, Ron Ferguson and Michael J. Mossinghoff
Journal: Math. Comp. 77 (2008), 1681-1694
MSC (2000): Primary 11Y35; Secondary 11M26
Published electronically: January 25, 2008
MathSciNet review: 2398787
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Abstract: The Liouville function $ \lambda(n)$ is the completely multiplicative function whose value is $ -1$ at each prime. We develop some algorithms for computing the sum $ T(n)=\sum_{k=1}^n \lambda(k)/k$, and use these methods to determine the smallest positive integer $ n$ where $ T(n)<0$. This answers a question originating in some work of Turán, who linked the behavior of $ T(n)$ to questions about the Riemann zeta function. We also study the problem of evaluating Pólya's sum $ L(n)=\sum_{k=1}^n\lambda(k)$, and we determine some new local extrema for this function, including some new positive values.


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Additional Information

Peter Borwein
Affiliation: Department of Mathematics and Statistics, Simon Fraser University, Burnaby, B.C. V5A 1S6 Canada
Email: pborwein@cecm.sfu.ca

Ron Ferguson
Affiliation: Department of Mathematics and Statistics, Simon Fraser University, Burnaby, B.C. V5A 1S6 Canada
Email: rferguson@pims.math.ca

Michael J. Mossinghoff
Affiliation: Department of Mathematics, Davidson College, Davidson, North Carolina 28035-6996
Email: mimossinghoff@davidson.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-08-02036-X
PII: S 0025-5718(08)02036-X
Keywords: Liouville function, P\'olya's sum, Tur\'an's sum, Riemann hypothesis
Received by editor(s): July 7, 2006
Published electronically: January 25, 2008
Additional Notes: The research of P. Borwein was supported in part by NSERC of Canada and MITACS
Article copyright: © Copyright 2008 American Mathematical Society