# Cycles in mathematics

## Numbers as cycles

a few notes

### Complex numbers

The introduction of complex numbers meant great progress - not only for solving of algebraic equations.
However, the whole issue is far from clear and transparent, note some pitfalls:

• when solving the cubic equation, a real root can be -in some cases- expressed only using roots of complex expressions (Casus irreducibilis)
• the possibility of displaying numbers in a complex (Gaussian) plane does not mean that, the both axes are "equivalent" (interchangeable).
• using the imaginary number "i" in the exponent does not create an exponential, but a periodic function: eix = cos x + i.sin x:
eix = 1 + i.x/1! - x2/2! - i.x3/3! + x4/4! ...
where cos x = 1 - x2/2! + x4/4! ... and sin x = x/1! - x3/3! ...
• the logarithm of the complex number is (in general) a multi-valued function
• failed to generalize complex numbers to multi-component numbers (satisfactorily)

### Euler's function Phi

The Euler's function of the number a = Π(pi) is defined by:
φ(a) = a. Π(1-1/pi), where pi are prime numbers of decomposition.
We can rewrite this expression to form:

#### φ(a) = a / Π(1, pi)

,

where expression (1, p) is synodical period of prime p with regard to number 1.

### Riemann's function Zeta

The Riemann's function Zeta is defined by:
ζ(s) = Π(1-1/pi)-s where pi are all prime numbers.
We can rewrite this expression to form:

#### ζ(s) = Π(1, pis)

,

where expression (1, ps) is synodical period of prime power ps with regard to number 1.

### Functions Eta versus Zeta

The Dirichlet's function Eta is defined by:
η(s) = (1-1/2s-1) * ζ(s).
We can rewrite this expression to form:

#### η(s) = ζ(s) * (1, 2s-1))

,

where expression (1, 2s-1) is synodical period of (s-1)-th power of the number 2 with regard to number 1.

### Mertens theorem

For n going to infinity:

#### Π(1, pi) = eλ . ln(pi)

,

where expression (1, pi) is synodical period of prime p with regard to number 1.