@GerhardPaseman Sorry if I am misunderstanding, but does this approach lead to an expression for $c(\theta)?$ Or why the leading term must be of this form?

So this suggests the answer should involve generalized harmonic numbers in some way. How is not entirely clear to me.

I can easily see that this sum looks as follows: for $n$ less than or equal to $x$ but strictly larger than $x/2,$ the summands are $1.$ For $x/3< n \leq x/2$ they are $2^{-\theta}$ and so on. But I am having trouble using this to get anywhere

@HarryRichman It is from the book ”A course in analytic number theory” by Marius Overholt.

@GerhardPaseman It is obvious to me that most of the summands are between $0$ and $1$. Does this observation make the answer straightforward?