I'm working through the problems in Montgomery & Vaughan's Multiplicative Number Theory. In Section 11.2 'Exceptional Zeros', Exercise 9a says that for a quadratic character $\chi$, show that for all $k\ge 0, x\ge1 $ $ \sum_{n where $\lambda$ is Liouville's function. This is elementary. In part b, under the hypothesis that there exists a $k$ such that $ \sum_{n for all $x\ge 1$, (no such $k$ is known to exist), one is to show that for all quadratic $\chi$ and all $\sigma>0$ $ L(\sigma,\chi)>0. $ I expect one is meant to use a Mellin transform with Cesàro weighting, S 5.1 in M&V. The difficulty is that $\chi(n)/n$ are the Dirichlet series coefficients of $L(s+1,\chi)$, not $L(s,\chi)$. Thus (5.18) gives $ L(\sigma+1,\chi)>0 $ for all $\sigma>0$.
Am I missing something obvious? Alternately, the method of part a will show that $ \sum_{n so under the hypothesis that there exists a $k$ such that for all $x\ge1$ $ \sum_{n one gets $L(\sigma,\chi)>0$ for all quadratic $\chi$.
EDIT: The reason one cares which version of part a is used, is that the numerics for small $k$ and moderate $x$ indicate that positivity is at least plausible for the original part a. It is not for the revised version.
Of course, one expects that no such $k$ exists in either case. For $k=0$, the positivity of $\sum_{n was known as Turán's conjecture, the negativity of $\sum_{n was Pólya's conjecture. Both where disproved by Haselgrove, and the same methods should generalize for any $k$.