  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.
