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Sign Changes of the Liouville Function on Quadratics
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- Journal:
- Canadian Mathematical Bulletin / Volume 56 / Issue 2 / 01 June 2013
- Published online by Cambridge University Press:
- 20 November 2018, pp. 251-257
- Print publication:
- 01 June 2013
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- Article
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- You have access
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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 
$bg{{\left( x \right)}^{2}}$.
BETWEEN THE PROBLEMS OF PÓLYA AND TURÁN
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 93 / Issue 1-2 / October 2012
- Published online by Cambridge University Press:
- 27 September 2012, pp. 157-171
- Print publication:
- October 2012
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- Article
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- You have access
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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.
