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: e^ix = cos x + i.sin x:
  e^ix = 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(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:

  φ(a) = a / Π(1, p_i)  

,

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/p_i)^-s where p_i are all prime numbers.
We can rewrite this expression to form:

  ζ(s) = Π(1, p_i^s)  

,

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

Functions Eta versus Zeta

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

  η(s) = ζ(s) * (1, 2^s-1))  

,

where expression (1, 2^s-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, p_i) = e^λ . ln(p_i)  

,

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