How the Superposition of the Periodic Pulsations of +1 and -1 Generates Values of the Mertens Function
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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 .
Contributed by: R. M. Abrarov and S. M. Abrarov (March 2011)
Open content licensed under CC BY-NC-SA
R. M. Abrarov and S. M. Abrarov, "Formulas for Positive, Negative and Zero Values of the Möbius Function," arXiv, 2009.