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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Completely multiplicative functions taking values in $ \{-1,1\}$


Authors: Peter Borwein, Stephen K. K. Choi and Michael Coons
Journal: Trans. Amer. Math. Soc. 362 (2010), 6279-6291
MSC (2000): Primary 11N25, 11N37; Secondary 11A15
Published electronically: July 14, 2010
MathSciNet review: 2678974
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Abstract: Define the Liouville function for $ A$, a subset of the primes $ P$, by $ \lambda_{A}(n) =(-1)^{\Omega_A(n)}$, where $ \Omega_A(n)$ is the number of prime factors of $ n$ coming from $ A$ counting multiplicity. For the traditional Liouville function, $ A$ is the set of all primes. Denote

$\displaystyle L_A(x):=\sum_{n\leq x}\lambda_A(n)\quad\mbox{and}\quad R_A:=\lim_{n\to\infty}\frac{L_A(n)}{n}.$

It is known that for each $ \alpha\in[0,1]$ there is an $ A\subset P$ such that $ R_A=\alpha$. Given certain restrictions on the sifting density of $ A$, asymptotic estimates for $ \sum_{n\leq x}\lambda_A(n)$ can be given. With further restrictions, more can be said. For an odd prime $ p$, define the character-like function $ \lambda_p$ as $ \lambda_p(pk+i)=(i/p)$ for $ i=1,\ldots,p-1$ and $ k\geq 0$, and $ \lambda_p(p)=1$, where $ (i/p)$ is the Legendre symbol (for example, $ \lambda_3$ is defined by $ \lambda_3(3k+1)=1$, $ \lambda_3(3k+2)=-1$ ($ k\geq 0$) and $ \lambda_3(3)=1$). For the partial sums of character-like functions we give exact values and asymptotics; in particular, we prove the following theorem.


Theorem.

If $ p$ is an odd prime, then

$\displaystyle \max_{n\leq x} \left\vert\sum_{k\leq n}\lambda_p(k)\right\vert \asymp\log x.$

This result is related to a question of Erdős concerning the existence of bounds for number-theoretic functions. Within the course of discussion, the ratio $ \phi(n)/\sigma(n)$ is considered.


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Additional Information

Peter Borwein
Affiliation: Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
Email: pborwein@cecm.sfu.ca

Stephen K. K. Choi
Affiliation: Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
Email: kkchoi@math.sfu.ca

Michael Coons
Affiliation: The Fields Institute, 222 College Street, Toronto, Ontario, Canada M5T 3J1
Email: mcoons@math.uwaterloo.ca

DOI: http://dx.doi.org/10.1090/S0002-9947-2010-05235-3
PII: S 0002-9947(2010)05235-3
Keywords: Liouville lambda function, multiplicative functions
Received by editor(s): June 13, 2008
Published electronically: July 14, 2010
Additional Notes: This research was supported in part by grants from NSERC of Canada and MITACS
Article copyright: © Copyright 2010 by the authors