Shrouded in mist lies the mysterious Möbius function. Unlike any usual curve which exhibits regular and smooth behavior, no one can tell what will the Möbius function will do next.
This lasted for a long time until August Ferdinand Möbius managed to give some meaning to the function, which incidentally was named after him. Let’s learn about this exotic function.
MöBIUS FUNCTION >
INTRODUCTION TO MöBIUS FUNCTION
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HAT 0
FUNDAMENTAL THEOREM OF ARITHMETIC
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VIDEO LECTURE
MAIN CONCEPTS
Hat 0, integers which are multiples of square numbers. Hat -1, integers that factors into an odd number of distinct primes. Hat +1, integers that factors into an even number of distinct primes. Get the Printer Friendly Version COMMENTS
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CONCEPT
The Möbius function is represented by the Greek letter
 , pronounced as mu. It is a function that sorts out positive integers into 3 different groups. To make things a little mystical, we shall call these groups hats.
The first hat is labeled “0”, the second “-1” and the third “+1” Here goes how the positive integers are distributed.
In Hat 0, Möbius places multiples of square numbers, other than 1 into this hat. The list goes like {4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32,36, …}. A square number is one which can be written in the form Before looking into the other two hats, we need to understand a theorem. The fundamental theorem of arithmetic says that every positive integer factors into a unique set of prime numbers, less the order they are written. A proof of this theorem is given in the next section. Keeping that in mind, Möbius places any integer that factors into an odd number of distinct primes into Hat -1. This includes {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 30, 31, 37, 41, 42, …}. An example would be say Finally, looking into Hat +1, we see numbers that factor into an even number of distinct primes, this time including 1 for completeness. Numbers in this hat includes {1, 6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39, 46, 51, 55, …}. The number 33 is in this hat because We can form a distinction between Hat 0 and Hats +1 and -1. Hats +1 and -1 contain integers where its factors are unique. This is unlike those integers in Hat 0, integers which have perfect squares as its factors. See, so long a square number is one of the factors, it can in effect be written as two identical numbers multiplied by itself, i.e. This should be a good enough introduction to the Möbius function. Now, let us look deeper in Hat 0.
All information presentated, less questions and exercises, is original content of Donny, with slight references to various books.
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namely 4, 9, 16, or 25 and this hat contains multiplies of such numbers. For example 
but 
.
because 
which is the only way 42 can be factorized into a set of prime numbers and there is an odd number of factors. Also notice that the list starts with prime numbers which must be the case where their factor is themselves, a single factor which is odd.
, where there are 2 distinct factors.
, which is already not unique. Another term for these integers is ‘non square free’ Basically, they cannot be freed from a square number being one of their factors as defined by Möbius.
