If the sum of all natural numbers that seems to be has a definite value, it is out of our common sense and it is difficult to take it for granted, even if you have some numerical sense. Proving that the series expansion is really mathematically correct no matter what value is derived from the diverging series is another problem, because it is too difficult to prove a series without convergent values that are initially assumed. Ramanujan expanded an infinite series of genius in this assumption and added a more mysterious interpretation.

Greek Letters
The definition of functions

Negative Domain
1. Polynomial expansion
2. Negative gamma function relation
3. Expansion of negative gamma function relation
4. Series of trigonometric and hyperbolic functions
5. Gamma function relation of subordinate zeta function
6. Coefficient of the subordinate zeta function

Positive Domain
1. Sine and cosine infinite series
2. Integral of sine and cosine infinite series
3. Sum of sine and cosine infinite series
4. L-function and subordinate zeta functions
5. Integral constant of sine and cosine infinite series
6. Proof of transcendental number

Analytic Continuation
1. Complex analysis
2. Functional equations
3. Riemann hypothesis

Appendix A. Gamma function formulas
Appendix B. Cotangent series and zeta function