The Möbius function

is defined for positive integers by

,

, where

is the number of prime factors of

if

is square-free, and

if

is not square-free (a number is square-free if its prime decomposition contains no repeated factors). The Mertens function

is the cumulative sum of the Möbius function

,

. For all positive integers

and

with

, build the table

, where the resulting multiplication sign (≠0) indicates to add a pulse (+1) or subtract a pulse (-1), while the resulting absolute value (≠0)

gives the period of pulsation (the gap along the axis of real numbers between consecutive pulses). Each pulsation of +1 and -1 for a particular combination of

begins from the first integer number divisible by

in the interval

if

or

if

and goes on through the period

within the interval. The function

is the superposition of the periodic pulsations of +1 and -1 for all combinations of

in the green area of the table with corresponding period determined by the table. 0 in the table signifies that the corresponding combinations of

do not contribute in

. The initial value

is 2. The table shows all available pulsations for specific

that can be involved in the calculation of the

function (green and yellow areas in the table). The green area shows pulsations actually involved in the calculation of the

function. By manipulating indices

and

we can expand or narrow the green area. Index

indicates the maximum rank of the green area while index

is the parameter of the completion of this area until a perfect square (

,

). When all available pulsations are involved in the calculation of the

function (no yellow area in the table), we will get for

exactly the values of the Mertens function

.