Sign changes in sums of the Liouville function
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- by Peter Borwein, Ron Ferguson and Michael J. Mossinghoff PDF
- Math. Comp. 77 (2008), 1681-1694 Request permission
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.References
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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
- MR Author ID: 630072
- ORCID: 0000-0002-7983-5427
- Email: mimossinghoff@davidson.edu
- 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
- © Copyright 2008 American Mathematical Society
- Journal: Math. Comp. 77 (2008), 1681-1694
- MSC (2000): Primary 11Y35; Secondary 11M26
- DOI: https://doi.org/10.1090/S0025-5718-08-02036-X
- MathSciNet review: 2398787