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

Footnotes
References
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[1]Borwein P., Ferguson R. and Mossinghoff M. J., ‘Sign changes in sums of the Liouville function’, Math. Comp. 77(263) (2008), 16811694.
[2]GMP: The GNU multiple precision arithmetic library. http://gmplib.org.
[3]Haselgrove C. B., ‘A disproof of a conjecture of Pólya’, Mathematika 5 (1958), 141145.
[4]Ingham A. E., ‘On two conjectures in the theory of numbers’, Amer. J. Math. 64 (1942), 313319.
[5]Lehman R. S., ‘On Liouville’s function’, Math. Comp. 14 (1960), 311320.
[6]Murty M. R., Problems in Analytic Number Theory, 2nd edn, Graduate Texts in Mathematics, 206 (Springer, New York, 2008).
[7]Pólya G., ‘Verschiedene Bemerkungen zur Zahlentheorie’, Jahresber. Deutsch. Math.-Verein. 28 (1919), 3140.
[8]Rubinstein M. and Sarnak P., ‘Chebyshev’s bias’, Experiment. Math. 3(3) (1994), 173197.
[9]Tanaka M., ‘A numerical investigation on cumulative sum of the Liouville function’, Tokyo J. Math. 3(1) (1980), 187189.
[10]Titchmarsh E. C., The Theory of the Riemann Zeta-Function, 2nd edn (Oxford University Press, New York, 1986).
[11]Turán P., ‘On some approximative Dirichlet-polynomials in the theory of the zeta-function of Riemann’, Danske Vid. Selsk. Mat.-Fys. Medd. 24(17) (1948), 136.
[12]Turán P., ‘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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