What are the equivalent statements of GRH using the Möbius or Liouville functions? We all know that Riemann Hypothesis can be stated as properties of $\mu$ or $\lambda$, particularly in terms of the random behaviour of those functions with "square root" bounds.
Are there similar equivalent statements for GRH (Generalized Riemann Hypothesis) in term of Möbius or Liouville function ?  
I assume such statements exist, but I have not seen them in popular textbooks. 
Can anyone share some links or pointers ?
Thanks in advance.
 A: Well GRH is inherently about Dirichlet characters, so they should be present somehow. For example, we have that for $\Re(s) > 1$,
\[\sum_{n=1}^{\infty} \frac{\lambda(n) \chi(n)}{n^s} = \prod_p \sum_{k = 0}^{\infty} \frac{(-1)^k \chi(p)^k}{p^{ks}} = \prod_p \frac{1}{1 + \chi(p) p^{-s}} = \prod_p \frac{1 - \chi(p) p^{-s}}{1 - \chi^2(p) p^{-2s}} = \frac{L(2s,\chi^2)}{L(s,\chi)}.
\]
If GRH is true for $L(s,\chi)$, then the right-hand side is holomorphic for $\Re(s) > 1/2$, so by the usual methods, one can show that for all $\varepsilon > 0$,
\[\sum_{n \leq x} \lambda(n) \chi(n) \ll_{\varepsilon} x^{1/2 + \varepsilon}.
\]
Conversely, this upper bound holding for all $\varepsilon > 0$ implies that the left-hand side is holomorphic for all $\Re(s) > 1/2$, which implies GRH for $L(s,\chi)$.
Similarly,
\[\sum_{n = 1}^{\infty} \frac{\mu(n) \chi(n)}{n^s} = \prod_p \left(1 - \frac{\chi(p)}{p^s}\right) = \frac{1}{L(s,\chi)}
\]
for $\Re(s) > 1$, and if GRH is true for $L(s,\chi)$ then the right-hand side is holomorphic for $\Re(s) > 1/2$, which implies that
\[\sum_{n \leq x} \mu(n) \chi(n) \ll_{\varepsilon} x^{1/2 + \varepsilon}
\]
for all $\varepsilon > 0$. Again, the converse is also true.
One can also express this in terms of arithmetic progressions using the orthogonality relations for Dirichlet characters. If GRH is true for $L(s,\chi)$ for all Dirichlet characters $\chi$ modulo $q$, then for any $(a,q) = 1$,
\[\sum_{\substack{n \leq x \\ n \equiv a \pmod{q}}} \lambda(n) \ll_{\varepsilon} x^{1/2 + \varepsilon}, \qquad \sum_{\substack{n \leq x \\ n \equiv a \pmod{q}}} \mu(n) \ll_{\varepsilon} x^{1/2 + \varepsilon}
\]
for all $\varepsilon > 0$. Conversely, if
\[\sum_{\substack{n \leq x \\ n \equiv a \pmod{q}}} \lambda(n) \ll_{\varepsilon} x^{1/2 + \varepsilon}, \qquad \sum_{\substack{n \leq x \\ n \equiv a \pmod{q}}} \mu(n) \ll_{\varepsilon} x^{1/2 + \varepsilon}
\]
for all $\varepsilon > 0$ and all $(a,q) = 1$, then GRH holds for $L(s,\chi)$ for any Dirichlet characters modulo $q$.
