We know that the prime number theorem is equivalent to the statement $$ M(x)=\sum_{n\le x}\mu(n)=o(x). $$ By using Ramanujan sums, we can write $M(x)$ as $$ M(x)=\sum_{q\le x}\sum_{\substack{0\lt a\lt q \\ (a,q)=1}} e\Big(\frac{a}{q}\Big), $$ and this resembles some sums that appear in the proof of the large sieve. For example, we know that for any 1-periodic function $f$ with continuous derivative, we have $$ \sum_{q\le x}\sum_{\substack{0\lt a\lt q \\ (a,q)=1}}\Big|f\Big(\frac{a}{q}\Big)\Big|\le x^2\int_{0}^{1}|f(t)|dt + \int_{0}^{1} |f'(t)|dt. $$ Is there any way to adapt this type of machinery for $M(x)$?

We know that the prime number theorem is equivalent to the statement $$ M(x)=\sum_{n\le x}\mu(n)=o(x). $$ By using Ramanujan sums, we can write $M(x)$ as $$ M(x)=\sum_{q\le x}\sum_{\substack{0\lt a\le q \\ (a,q)=1}} e\Big(\frac{a}{q}\Big), $$ and this resembles some sums that appear in the proof of the large sieve. For example, we know that for any 1-periodic function $f$ with continuous derivative, we have $$ \sum_{q\le x}\sum_{\substack{0\lt a\lt q \\ (a,q)=1}}\Big|f\Big(\frac{a}{q}\Big)\Big|\le x^2\int_{0}^{1}|f(t)|dt + \int_{0}^{1} |f'(t)|dt. $$ Is there any way to adapt this type of machinery for $M(x)$?