Analytic number theory is the study of number theory using techniques from analysis, most notably complex analysis (essentially, calculus done with complex numbers).
On its face, this seems like a completely crazy idea: analysis works with smooth functions, yet in number theory, we aren't typically interested in smoothness or even continuity---typically, we want answers for the integers!
Nevertheless, it turns out that many number theoretic functions can be approximated by smooth functions---figuring out exactly what and how good these approximations are is a big part of the theory.
Another approach that you can take is to take a number theoretic function, build a nice smooth function out of it (classically, an L-function or an automorphic form or a mock modular form---at this point, there is a whole zoo of these things), and study this function. If you are lucky, by studying this new function closely, you can learn things about your original number theoretic function.
Perhaps an example is in order. Consider the partition function [math] p(n) [/math], defined to be the number of ways that you can write the integer [math] n [/math] as a sum of positive integers (without regard to order). So, for example, [math] p(3) = 3 [/math] because:
[math] 3 = 3 [/math]
[math] 3 = 2 + 1 [/math]
[math] 3 = 1 + 1 + 1 [/math]
A good way to study the partition function is to instead consider the smooth function [math] G(q) = \sum_{n = 0}^\infty p(n) q^n [/math], which is called the generating function of [math] p(n) [/math]. You can get beautiful results this way, including the following remarkable exact formula for the partition number:
where:
where:
[math] s(m, k) = \sum_{n \mod k} \left(\left(\frac{n}{k}\right)\right)\left(\left(\frac{mn}{k}\right)\right) [/math]
where:
[math] \left(\left(x\right)\right) = \begin{cases} x - \lfloor x \rfloor - \frac{1}{2} & \text{if } x \notin \mathbb{Z} \\ 0 & \text{if } x \in \mathbb{Z} \end{cases} [/math]
(I said that it was remarkable, not that it was easy to write down.)
On its face, this seems like a completely crazy idea: analysis works with smooth functions, yet in number theory, we aren't typically interested in smoothness or even continuity---typically, we want answers for the integers!
Nevertheless, it turns out that many number theoretic functions can be approximated by smooth functions---figuring out exactly what and how good these approximations are is a big part of the theory.
Another approach that you can take is to take a number theoretic function, build a nice smooth function out of it (classically, an L-function or an automorphic form or a mock modular form---at this point, there is a whole zoo of these things), and study this function. If you are lucky, by studying this new function closely, you can learn things about your original number theoretic function.
Perhaps an example is in order. Consider the partition function [math] p(n) [/math], defined to be the number of ways that you can write the integer [math] n [/math] as a sum of positive integers (without regard to order). So, for example, [math] p(3) = 3 [/math] because:
[math] 3 = 3 [/math]
[math] 3 = 2 + 1 [/math]
[math] 3 = 1 + 1 + 1 [/math]
A good way to study the partition function is to instead consider the smooth function [math] G(q) = \sum_{n = 0}^\infty p(n) q^n [/math], which is called the generating function of [math] p(n) [/math]. You can get beautiful results this way, including the following remarkable exact formula for the partition number:
where:
[math] s(m, k) = \sum_{n \mod k} \left(\left(\frac{n}{k}\right)\right)\left(\left(\frac{mn}{k}\right)\right) [/math]
where:
[math] \left(\left(x\right)\right) = \begin{cases} x - \lfloor x \rfloor - \frac{1}{2} & \text{if } x \notin \mathbb{Z} \\ 0 & \text{if } x \in \mathbb{Z} \end{cases} [/math]
(I said that it was remarkable, not that it was easy to write down.)
About the Author
Senia Sheydvasser
Math PhD Student at Yale UniversityWorks at Yale University
Studied at Bowdoin College
Lives in New Haven, CT
10.2m answer views465.5k this month
Top Writer2017 and 2016