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:

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 jis 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.