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BETWEEN THE PROBLEMS OF PÓLYA AND TURÁN

  • MICHAEL J. MOSSINGHOFF (a1) and TIMOTHY S. TRUDGIAN (a2)
Abstract
Abstract

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.

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For correspondence; e-mail: mimossinghoff@davidson.edu
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Dedicated to the memory of Alf van der Poorten

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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[1] P. Borwein , R. Ferguson and M. J. Mossinghoff , ‘Sign changes in sums of the Liouville function’, Math. Comp. 77(263) (2008), 16811694.

[4] A. E. Ingham , ‘On two conjectures in the theory of numbers’, Amer. J. Math. 64 (1942), 313319.

[5] R. S. Lehman , ‘On Liouville’s function’, Math. Comp. 14 (1960), 311320.

[8] M. Rubinstein and P. Sarnak , ‘Chebyshev’s bias’, Experiment. Math. 3(3) (1994), 173197.

[9] M. Tanaka , ‘A numerical investigation on cumulative sum of the Liouville function’, Tokyo J. Math. 3(1) (1980), 187189.

[12] P. Turán , ‘Nachtrag zu meiner Abhandlung “On some approximative Dirichlet polynomials in the theory of zeta-function of Riemann”’, Acta Math. Acad. Sci. Hungar. 10 (1959), 277298.

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Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
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