Due to the irregularity of the Möbius Function, mathematicians found it exceeding difficult to analyze it. They then decided to define another function called the Mertens function which is the cumulative sum of the Möbius function.
This Mertens function has a colorful history of conjectures, some of which are still very much alive even till today.
MöBIUS FUNCTION >
MERTENS FUNCTION
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UNIQUENESS OF FACTORIZATION
MERTENS CONJECTURE
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1977 AHSME #16
VIDEO LECTURE
MAIN CONCEPTS
The Mertens function,  , is the cumulative sum of the Möbius function up to the x term. Get the Printer Friendly Version COMMENTS
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CONCEPT
When we list the first twenty terms of the Möbius function, we have the following series:
or illustrated on the following graph
The graph gives us a chaotic pattern and no meaningful analysis can be drawn from it. Mathematicians then decided to see what they get when they took the cumulative sum of the Möbius function. That is
which was latter designated as the Mertens function, or
Now, we are getting somewhere. At least the graph is taking some shape. The irregularity of the graph still exists but at least a pattern can be suspected. I also would like to point out that some graphs taking such unusual shapes can be actually modeled after an algebraic function, the case here is the Mertens function. If your friend tells you to find a function that draws out a chaotic graph, don’t be alarmed. It could exist! All information presentated, less questions and exercises, is original content of Donny, with slight references to various books.
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