An investigation of the asymptotic behavior of the Mertens function

It was believed that the Mertens function is a simple random walk in the first versions of the article, so its asymptotic behavior obeys the law of the iterated logarithm. In the latest version of the article we show why the asymptotic estimate – the law of the iterated logarithm – does not hold for the Mertens function. A refined asymptotic estimate for the Mertens function is proved.