Proper Way To Compute An Upper Bound

I regard to the proof of Lemma 10 in “A remark on a conjecture of Chowla” by M. R. Murty, A. Vatwani, J. Ramanujan Math. Soc., 33, No. 2, 2018, 111-123,

the authors used the average value $ (\log x)^c$ , $ c$ constant, of the number of divisors function $ \tau(d)=\sum_{d|n}1$ as an upper bound for $ \tau(d)^2$ , where $ d \leq x$ . To be specific, they claim that $ $ \sum_{q \leq x^{2\delta}}\tau(q)^2 \left | \sum_{\substack{m \leq x+2\ m \equiv a \bmod q}} \mu(m)\right | \ll x (\log x)^{2c},$ $

where $ 2 \delta <1/2$ .

The questions are these:

  1. Is the main result invalid? The upper bound should be $ $ \sum_{q \leq x^{2\delta}}\tau(q)^2 \left | \sum_{\substack{m \leq x+2\ m \equiv a \bmod q}} \mu(m)\right | \ll x ^{1+2\delta}.$ $ This is the best unconditional upper bound, under any known result, including Proposition 3.

  2. It is true that the proper upper bound $ \tau(d)^2 \ll x^{2\epsilon}$ , $ \epsilon >0$ , is not required here?

  3. Can we use this as a precedent to prove other upper bounds in mathematics?