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

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

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

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.