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 $
$$
\sum_{n
Am I missing something obvious? Alternately, the method of part a will show that
$$
\sum_{n
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