BETWEEN THE PROBLEMS OF PÓLYA AND TURÁN

Journal of the Australian Mathematical Society

Journal of the Australian Mathematical Society / Volume 93 / Issue 1-2 / October 2012, pp 157-171
Copyright © 2012 Australian Mathematical Publishing Association Inc.
DOI: http://dx.doi.org/10.1017/S1446788712000201 (About DOI), Published online: 27 September 2012

Table of Contents - October 2012 - Volume 93, Issue 1-2

Research Article

BETWEEN THE PROBLEMS OF PÓLYA AND TURÁN

Article author query
mossinghoff mj [Google Scholar]
trudgian ts [Google Scholar]

MICHAEL J. MOSSINGHOFF^a1 ^c1 and TIMOTHY S. TRUDGIAN^a2

^a1Department of Mathematics, Davidson College, Davidson, NC, 28035-6996, USA (email: mimossinghoff@davidson.edu)

^a2Department of Mathematics and Computer Science, University of Lethbridge, Alberta, Canada T1K 3M4 (email: tim.trudgian@uleth.ca)

Abstract

We investigate the behaviour of the function $L_{\alpha }(x) = \sum _{n\leq x}\lambda (n)/n^{\alpha }$ \$L_{\alpha }(x) = \sum _{n\leq x}\lambda (n)/n^{\alpha }\$, where $\lambda (n)$ \$\lambda (n)\$ is the Liouville function and $\alpha$ \$\alpha \$ is a real parameter. The case where $\alpha =0$ \$\alpha =0\$ was investigated by Pólya; the case $\alpha =1$ \$\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 $L_{\alpha }(x)$ \$L_{\alpha }(x)\$, where $0\leq \alpha \leq 1$ \$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 $\alpha =1/2$ \$\alpha =1/2\$is of particular interest.