Yeah, there’s definitely a few relevant things here:
Representation theory is relevant (in particular representations of cyclic groups, which is basically the circle arithmetic you’re talking about). Representation theory gives you matrices that use complex numbers, even for discrete finite groups. So number theory isn’t the only place where complex numbers poke their nose into discrete business.
There’s apparently a whole theory of Dirichlet series and Dirichlet convolution which is analogous to Fourier series and Fourier convolution. Complex numbers are the nicest way to do Fourier series, so it makes sense that they’re also the nicest way to do Dirichlet stuff.
I guess just in general complex numbers are the best field and a lot of the math of number theory is turning things into vector spaces and linear operators and doing linear algebra on them. And if you’re picking a field for your vector space, what better choice than C?
A Dirichlet series is a function of a variable s, and complex functions can have complex analysis done on them, and complex analysis is uniquely nice, so why not make s a complex variable?
The post author seems to already know a lot of math, so I guess they’re looking for a deeper kind of answer.
Yeah, there’s definitely a few relevant things here:
Representation theory is relevant (in particular representations of cyclic groups, which is basically the circle arithmetic you’re talking about). Representation theory gives you matrices that use complex numbers, even for discrete finite groups. So number theory isn’t the only place where complex numbers poke their nose into discrete business.
There’s apparently a whole theory of Dirichlet series and Dirichlet convolution which is analogous to Fourier series and Fourier convolution. Complex numbers are the nicest way to do Fourier series, so it makes sense that they’re also the nicest way to do Dirichlet stuff.
I guess just in general complex numbers are the best field and a lot of the math of number theory is turning things into vector spaces and linear operators and doing linear algebra on them. And if you’re picking a field for your vector space, what better choice than C?
A Dirichlet series is a function of a variable s, and complex functions can have complex analysis done on them, and complex analysis is uniquely nice, so why not make s a complex variable?
The post author seems to already know a lot of math, so I guess they’re looking for a deeper kind of answer.