A special case of a well known result of Ingham is that

$$\sum_{n\leq x} d(n)d(n+1)=\frac{6}{\pi^2}x(\log x)^2+O(x\log x)$$

where $d(n)$ is the number of divisors of $n$.

Ingham's results, which are arrived at by a long but elementary calculation, may be combined with a further long and non-elementary (i.e. involving some complex analysis) calculation, but which does not use any properties of non-principal characters or $L$-functions, to arrive at the conclusion that the following average over all $q$ less than $x $ of all non-principal character sums:

$$\frac{1}{x}\sum_{q\leq x}\frac{1}{\phi(q)}\sum_{\chi\neq \chi_0}\left(\sum_{n\leq x}\chi(n)d(n)\right)=O(\log x).$$

In fact, it can be shown that this statement is also sufficient for the truth of Ingham's result.

It seems to me that statements like this would have more direct proofs, perhaps analytical using known or hypothetical properties of $L^2(s,\chi)$, or perhaps using properties of character sums. Is this so?