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Sign Changes of the Liouville Function on Quadratics

Published online by Cambridge University Press:  20 November 2018

Peter Borwein ,
Stephen K. K. Choi  and
Himadri Ganguli
Peter Borwein
Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby, BC, V5C 1S6 e-mail: pborwein@sfu.cakkchoi@math.sfu.cahganguli@sfu.ca
Stephen K. K. Choi
Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby, BC, V5C 1S6 e-mail: pborwein@sfu.cakkchoi@math.sfu.cahganguli@sfu.ca
Himadri Ganguli
Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby, BC, V5C 1S6 e-mail: pborwein@sfu.cakkchoi@math.sfu.cahganguli@sfu.ca
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Abstract

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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 $bg{{\left( x \right)}^{2}}$.

Keywords

11N6011B8311D09Liouville functionChowla's conjectureprime number theorembinary sequenceschanges sign infinitely oftenquadratic polynomialsPell equations
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 56 , Issue 2 , 01 June 2013 , pp. 251 - 257
Copyright
Copyright © Canadian Mathematical Society 2013

References

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