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

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 \[ L_A(x):=\sum _{n\leq x}\lambda _A(n)\quad \mbox {and}\quad R_A:=\lim _{n\to \infty }\frac {L_A(n)}{n}.\] 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 \[ \max _{n\leq x} \left |\sum _{k\leq n}\lambda _p(k)\right | \asymp \log x.\]

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.