Mobius Function - page 1

1. Introduction.
I have completely extended Analytic Number Theory by applying it to complex numbers. I ask the reader to put their prior knowledge of complex numbers aside.
Form an early age we have been taught that complex numbers can have no real meaning and that complex forces exist but without any evidence that this is true.

I recommend the book "An Introduction to Analytic Number Theory" by Apostol (an O.U. set book).

In Analytic Number Theory the prime numbers follow a logarithmic distribution and in my definition of complex space distances are logarithmic. Therefore the possibility of having complex waveforms following prime number patterns arises. As a consequence a waveform can represent a series of vectors so this also establishes the possibility of having complex forces and telling us something about their behaviour.
I also define the complex plane has having one face in 3D (the interior face) and having one face in 4D (the exterior face).

The Mobius Function
I am inspecting functions that Analytic Number Theory is based upon because they demonstrate some hidden properties or pre-requisites for establishing a valid and real complex space. These assumptions have been made unwittingly but are nevertheless very significant.

1. I note that the sum of the (n) numbers that are squared in the Mobius Function is zero. Interpret this as laying down a basis for establishing a directionality between real 3D space with (n^2) numbers and complex space with (n) numbers.

2. The Mobius Function for the numbers 4 and 8 is zero for both because they have a factor of 2^2=4.

See "Table for Mobius Function"

3. Powers of primes are always 1 for the Mobius Function to equal -1. This suggests to me that the theory has incorporated features that show the nature of complex space. In this case that when the function equals -1 they function touches the complex plane and is reminiscent of i^2 = -1

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