a few notes

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:
e
^{ix}= cos x + i.sin x:

e^{ix}= 1 + i.x/1! - x^{2}/2! - i.x^{3}/3! + x^{4}/4! ...

where cos x = 1 - x^{2}/2! + x^{4}/4! ... and sin x = x/1! - x^{3}/3! ... - the logarithm of the complex number is (in general) a multi-valued function
- failed to generalize complex numbers to multi-component numbers (satisfactorily)

The Euler's function of the number a = Π(p_{i}) is defined by:

φ(a) = a. Π(1-1/p_{i}), where p_{i} are prime numbers of decomposition.

We can rewrite this expression to form:

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

The Riemann's function Zeta is defined by:

ζ(s) = Π(1-1/p_{i})^{-s} where p_{i} are all prime numbers.

We can rewrite this expression to form:

where expression (1, p^{s}) is synodical period of prime power p^{s} with regard to number 1.

The Dirichlet's function Eta is defined by:

η(s) = (1-1/2^{s-1}) * ζ(s).

We can rewrite this expression to form:

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

For n going to infinity:

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