I am searching for an introductory book in the field of complex analysis for self learning, that would contain the following:

- Analytic number theory : the connection between complex analysis and number theory and most importantly the Riemann hypothesis
- Some proofs : I prefer to look behind the proofs of the theorem that I study before accepting it fully
- The book must be suitable for self study, so it would be appreciated if it is clear enough and contains everything needed
- The book must contain the necessary parts of complex analysis that are used in physics, given that I'm interested in physics too
- I would like the algebraic approach of complex analysis but it would be good if the book contained the geometric beauty of complex numbers too like fractals (Julia set, Mandelbrot set)

I will use complex analysis later in two mathematical fields: Partial Differential Equations and Number Theory, so I am searching for a book that covers the fundamentals of complex analysis on an undergraduate level that will make me comfortable in studying PDE and Number Theory later on.

My background in mathematics: ODE - Real analysis - Multivariable analysis - Vector Analysis - Linear algebra (matrices and vector space)