Euler Totient Function 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.
From 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 Euler Totient Function
I am beginning with the underlying concepts in Analytic Number Theory to show how strongly connected my concept of complex space, waveforms, and forces correlates to underpin my ideas mathematically. If you wish to see what these complex waves look like you may jump ahead to later topics as I will provide illustrations. Alternatively you can download some first notes at my yahoo group.

The Euler Totient function of a number counts the quantity of relatively prime numbers that are smaller to the original number. So any number with a common factor is not relatively prime (including one).

In figure Euler Totient 1 a table of values for the function is given:

In figure Euler Totient 2 a start is made to deviate from normal practice.
 
The function is shown graphically but following Prime Number Theory there is no need for this approach, just a consideration of the numbers themselves as progressions and behaviours. I am showing that basic ‘arithmetic’ functions like this underpin the concept of waves among the prime numbers and then I can associate the primes with complex numbers.

continued on page 2

In figure Euler Totient 3 we can say a pattern is beginning to emerge (that I expect will continue).

This pattern shown graphically is not a waveform. It is too early to introduce them. What this does is show that we have a function bounded exponentially. So if complex space has logarithmic distances as I expect this is relevant as a constraining device ( the logarithm of an exponential is a real number).

In figure Euler Totient Product 1a the possibility of linking the function to real numbers is stated mathematically.

 As I said this will come as a surprise to most mathematicians to see this approach. It may even be dismissed as an interesting distraction. However I will show that such connections exist throughout Analytic Number Theory so that the linkage with complex space becomes inescapable.

In figure Euler Totient Product 1b we can link these two functions together.

 Thus we are starting to establish the boundary conditions for complex waves ( I will so refer to these waves rather than keep citing “waves in Prime Number Theory”).
The Mobius Function only has values of -1, zero, and +1. This provides an opportunity to link it with the three classes of numbers: imaginary, those between real and complex space, and real numbers (as a first consideration).
More importantly linking the two functions together initiates the grouping and properties of numbers into separate classes (that I use uniquely).

In figure Euler Totient Product 1c there is an amazingly equation giving a product!

 The logarithm of a product (say ab) is the same as the sum of the logarithms for a and b. So I find this intriguing as it allows a link between an additive state (real space) and a multiplicative state (complex space).

In figure Euler Totient Product 1d

 The series generated reminds me of the Riemann Function and I believe it can represent action along the arm of a ztar ( a complex manifold I define).

In figure Euler Totient Product 2 there is a linear sinusoidal wave as distances on the complex plane are logarithmic converting the logarithmic distribution of the primes into a linear distribution and the resulting sinusoidal wave pattern.

 I associate each term with the amplitude (or area) of each phase. According to my theory the ztar has a geometry of y = 1/x. So the presence of factors which represent 1/p is significant and suggests the action is along the arm of a ztar although originating from the complex plane and into real space (the left hand side of the equation in figure Euler Totient Product 1d).