How the Superposition of the Periodic Pulsations of +1 and -1 Generates Values of the Mertens Function

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 .



  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


R. M. Abrarov and S. M. Abrarov, "Formulas for Positive, Negative and Zero Values of the Möbius Function," arXiv, 2009.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+