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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 $tau(d)=sum_n1$ as an upper bound for $tau(d)^2$, where $d leq x$. To be specific, they claim that
$$sum_q leq x^2deltatau(q)^2 left | sum_substackm leq x+2\
m equiv a bmod q mu(m)right | ll x (log x)^2c,$$

where $2 delta <1/2$.

The questions are these:

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




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




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

Good question, and I agree that the authors should have been more explicit here. However, I think I can reconstruct their argument: note that
beginalign*
sum_q leq x^2deltatau(q)^2 bigg | sum_substackm leq x+2\
m equiv a bmod q mu(m)bigg | &le sum_q leq x^2deltatau(q)^2 sum_substackm leq x+2\ m equiv a bmod q |mu(m)| \
&le sum_q leq x^2deltatau(q)^2 sum_substackm leq x+2\ m equiv a bmod q 1 \
&ll sum_q leq x^2deltatau(q)^2 frac xq = x sum_q leq x^2delta fractau(q)^2q.
endalign*
And this remaining sum is indeed $ll_delta (log x)^2c$ for some constant $c$; indeed, it's not hard to show that
$$
sum_q leq y fractau(q)^2q sim frac(log y)^44pi^2.
$$