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Let $\mu(k)$ the Möbius function and $M(n)$ the Mertens function, that is its summatory function $\sum_{1\leq k\leq n}\mu(k)$. And let $\operatorname{rad}(n)$ the squarefree kernel of $n$ or radical of an integer, see this Wikipedia. From these arithmetic functions we define for real numbers $x\geq 1$ the sequence $$\mathcal{A}(x):=\sum_{1\leq n\leq x}\frac{M(n)}{n\cdot\operatorname{rad}(n)}\log\left(\frac{x}{n}\right).\tag{1}$$

Question. I would like to know what claim can we get on assumption of any remarkable conjecture about the behaviour of previous arithmetic function, with the purpose to study the asymptotic behaviour of $\mathcal{A}(x)$ as $x\to\infty$. Many thanks.

As I've asked, I am looking conditional statements invoking remarkable and well-known conjectures in analytic nubmer theory, and if it is required to refer some unconditional statement in your approach/proof feel free to answer this question as a reference request adding to your final formula what statement of what article I should to read, and I try to search and to read such statement.

My unfinished approach. Assuming the Riemann hypothesis for each $\epsilon>0$ $$\mathcal{A}(x)=O\left(\int_1^x\frac{1}{t}\left(\sum_{1\leq n\leq t}\frac{n^{\epsilon-\frac{1}{2}}}{\operatorname{rad}(n)}\right)dt\right).$$ Here I've combined the well-known equivalence of the Riemann hypothesis related to the Mertens function the statement of Exercise 2.5.3 showed in page 29 of Ram Murty, Problems in Analytic Number Theory, Second Edition, Springer (2008), currently you can read a free view of this page from some edition of such book in Google Books.

My approach is unfinished because I should to combine with some additional conjecture, or the Riemann hypothesis again, providing me the behaviour of $$\sum_{1\leq n\leq y}\frac{n^{\epsilon-\frac{1}{2}}}{\operatorname{rad}(n)}$$ as $y\to\infty$.$\square$

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  • $\begingroup$ Thanks you @RenatoFaraone $\endgroup$ – user243301 Feb 23 '18 at 20:16
  • $\begingroup$ Have you even read what that tag is about ? $\endgroup$ – Renato Faraone Feb 23 '18 at 20:17
  • $\begingroup$ I don't know what is the full meaning of this tag, but if I was wrong in my choice of tags I hope help from this community. I add tags in my posts but maybe these aren't the most appropriate. Many thanks @RenatoFaraone $\endgroup$ – user243301 Feb 24 '18 at 11:16
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    $\begingroup$ "Summation method" is about ways of giving divergent series a meaningfull finite result in a generalized sense of sum, not about "ways I can sum or estimate a converging series" $\endgroup$ – Renato Faraone Feb 24 '18 at 11:20

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