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BETWEEN THE PROBLEMS OF PÓLYA AND TURÁN

Published online by Cambridge University Press:  27 September 2012

MICHAEL J. MOSSINGHOFF*
Affiliation:
Department of Mathematics, Davidson College, Davidson, NC, 28035-6996, USA (email: mimossinghoff@davidson.edu)
TIMOTHY S. TRUDGIAN
Affiliation:
Department of Mathematics and Computer Science, University of Lethbridge, Alberta, Canada T1K 3M4 (email: tim.trudgian@uleth.ca)
*
For correspondence; e-mail: mimossinghoff@davidson.edu
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Abstract

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We investigate the behaviour of the function $L_{\alpha }(x) = \sum _{n\leq x}\lambda (n)/n^{\alpha }$, where $\lambda (n)$ is the Liouville function and $\alpha $ is a real parameter. The case where $\alpha =0$ was investigated by Pólya; the case $\alpha =1$, by Turán. The question of the existence of sign changes in both of these cases is related to the Riemann hypothesis. Using both analytic and computational methods, we investigate similar problems for the more general family $L_{\alpha }(x)$, where $0\leq \alpha \leq 1$, and their relationship to the Riemann hypothesis and other properties of the zeros of the Riemann zeta function. The case where $\alpha =1/2$is of particular interest.

Type
Research Article
Copyright
Copyright © 2012 Australian Mathematical Publishing Association Inc.

Footnotes

Dedicated to the memory of Alf van der Poorten

References

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