Another estimating the absolute value of Mertens function

Abstract

Through an inversion approach, we suggest a possible estimation for the absolute value of Mertens function $|M(x)|$ that $\left|M(x)\right|\sim \left[\frac{1}{\pi \sqrt{\epsilon }(x+\epsilon )}\right]\sqrt{x}$ (where $x$ is an appropriately large real number, and $\epsilon$ ($0<\epsilon <1$) is a small real number which makes $2x+\epsilon$ to be an integer). For any large $x$, we can always find an $\epsilon$, so that $|M(x)|<\left[\frac{1}{\pi \sqrt{\epsilon }(x+\epsilon )}\right]\sqrt{x}$.

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