Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-12T19:45:11.882Z Has data issue: false hasContentIssue false

Sign Changes of the Liouville Function on Quadratics

Published online by Cambridge University Press:  20 November 2018

Peter Borwein
Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby, BC, V5C 1S6 e-mail: [email protected]@[email protected]
Stephen K. K. Choi
Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby, BC, V5C 1S6 e-mail: [email protected]@[email protected]
Himadri Ganguli
Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby, BC, V5C 1S6 e-mail: [email protected]@[email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let $\text{ }\!\!\lambda\!\!\text{ }\left( n \right)$ denote the Liouville function. Complementary to the prime number theorem, Chowla conjectured that

*

$$\sum\limits_{n\le x}{\lambda \,\left( f\left( n \right) \right)}=o\left( x \right)$$

for any polynomial $f\left( x \right)$ with integer coefficients which is not of form $bg{{\left( x \right)}^{2}}$.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2013

References

[1] Borwein, P., Ferguson, R., and Mossinghoff, M., Sign changes in sums of the Liouville function. Math. Comp. 77(2008), no. 263, 16811694. http://dx.doi.org/10.1090/S0025-5718-08-02036-X Google Scholar
[2] Cassaigne, J., Ferenczi, S., Mauduit, C., Rivat, J., and A. Sarkozy On finite pseudorandom binary sequences. IV. The Liouville function. II. Acta Arith. 95(2000), no. 4, 343359. Google Scholar
[3] Chowla, S., The Riemann Hypothesis and Hilbert's Tenth Problem. Gordon and Breach, New York, 1965.Google Scholar
[4] Haselgrove, C. B. A disproof of a conjecture of P´olya. Mathematika 5(1958), 141145. http://dx.doi.org/10.1112/S0025579300001480 Google Scholar
[5] Kátai, I., Research Problems in Number Theory. II. Ann. Univ. Sci. Budapest. Sect. Comput. 16(1996), 223251. Google Scholar
[6] Mollin, R., Central norms and continued fractions. Int. J. Pure Appl. Math. 55(2009), no. 1, 18. Google Scholar
[7] Mollin, R., Norm form equations and continued fractions. Acta Math. Univ. Comenian. 74(2005), no. 2, 273278. Google Scholar
[8] Mollin, R., Criteria for simultaneous solutions of X2 – DY2 = c and x2 – Dy2 = –c. Canad. Math. Bull. 45(2002), no. 3, 428435. http://dx.doi.org/10.4153/CMB-2002-045-2 Google Scholar
[9] Mollin, R., The Diophantine equation ax 2 – by 2 = c and simple continued fractions. Int. Math. J. 2(2002), no. 1, 16. Google Scholar
[10] Pólya, G., Verschiedene Bemerkungen zur Zahlentheorie. Jahresber. Deutsch. Math.-Verein. 28(1919), 3140. Google Scholar
[11] Sarkozy, A., Unsolved Problems in Number Theory. Period. Math. Hungar. 42 (2001), no. 1-2, 1735. http://dx.doi.org/10.1023/A:1015236305093 Google Scholar
[12] Siegel, C. L., Über einige Anwendungen Diophantischer Approximationen. Abh. Preussischen Akademie der Wissenshaften, Phys. Math. Klasse (1929) pp. 4169. Google Scholar