# 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?