Chapter 1. Introduction Chapter 2. Periodic patterns Chapter 3. Rationality of $\zeta _n$ Chapter 4. More symbols on larger lattice Chapter 5. Zeta functions presented in skew coordinates Chapter 6. Analyticity and meromorphic extensions of zeta functions Chapter 7. Equations on $\mathbb {Z}^2$ with numbers in a finite field Chapter 8. Square lattice Ising model with finite range interaction