Takuro Shintani studied zeta functions over totally real number fields. In 1976, he came upon a remarkable discovery, Shintani’s Unit Theorem. First, what is a totally real number field (you might even wish to know what a number field is, but I will save that for some future date). Fields over the rational numbers can be embedded into either the real numbers or the complex numbers (this is a consequence of the fundamental theorem of algebra).
An example of a totally real field is the field generated by adjoining a root of x2 - 2 to the rational numbers. Since there are two solutions to this polynomial, there are two embeddings. You send the root to √2 or -√2. Both of these are real numbers.
A nice consequence of this is that we can embed our field into a two-dimensional real vector space. This means that we now have linear algebra and vector calculus at our disposal. Another way to think about this is that the Minkowski Space is simple in some sense. I won’t go into what the Minkowski Space right now, but a good reference is sec. 3, chap. VII of Jürgen Neukirch’s Algebraic Number Theory (this is my favorite reference for algebraic number theory).
One way to examine number fields is through zeta functions. For totally real number fields, Shintani had a remarkable insight in that you can decompose the totally positive orthant in a nice way. This decomposition of the totally positive orthant is called Shintani’s Unit Theorem and it allowed Shintani to provide another proof on the rationality of zeta functions at non-positive integers.
I have swept such an enormous amount of details under the rug that you could hardly tell there ever was a rug. I hope to fill in the gaps in the future.