Mertens function

In number theory, the Mertens function is defined as:

M(n)=␡ ␡ 1≤ ≤ k≤ ≤ nμ μ (k){displaystyle M(n)=sum _{1leq kleq n}mu (k)}}

where μ(k) is the Möbius function. Since the Möbius function contemplates only the images {-1,0,1} it is obvious that the Mertens function hardly varies in its traversal and that there is no value of x for which |M (x)|>x. The Mertens conjecture goes further by stating that there is no value for x where the absolute value of the Mertens function exceeds the value of the square root of x, however, This conjecture has been shown to be false (if there are values of x such that the absolute value of the Mertens function is greater than the square root of x).

Some values of the Mertens function are 1, 0, -1, -1, -2, -1, -2, -2,... (sequence A002321 in OEIS).