It’s been a while since I’ve posted – I’m sorry. I’ve been busy, perhaps working on a paper (let’s hope it becomes a paper) and otherwise trying to learn things. This post is very closely related to some computations that have been coming up in what I’m currently looking at (in particular, looking at h-th coefficients of Eisenstein series of half-integral weight). I hope to write a very expository-level article on this project that I’ve been working on, outsourcing but completely providing computations behind the scenes in posts such as this one.
I’d like to add that this post took almost no time to write, as I used some vim macros and latex2wp to automatically convert a segment of something I’d written into wordpress-able html containing the latex. And that’s pretty awesome.
There is a particular calculation that I’ve had to do repeatedly recently, and that I will mention and use again. In an effort to have a readable account of this calculation, I present one such account here. Finally, I cannot help but say that this (and the next few posts, likely) are all joint work with Chan and Mehmet, also from Brown University.
Let us consider the following generalized Gauss Sum:
where I let be the Legendre Symbol, and there is the sign of the th Gauss sum, so that it is if and it is if . It is not defined for even.
Lemma 1 is multiplicative in .
Proof: Let be two relatively prime integers. Any integer can be written as , where runs through integers and runs with the Chinese Remainder Theorem. Then
Using quadratic reciprocity, we see that , so that .
Let’s calculate for prime powers . Let be a primitive th root of unity. First we deal with the case of odd , . If , we have the typical quadratic Gauss sum multiplied by
For , we will see that is . We split into cases when is even or odd. If is even, then we are just summing the primitive th roots of unity, which is . If is odd,
since the inner sum is again a sum of roots of unity. Thus
Notice that this matches up with the th part of the Euler product for .
Now consider those odd such that . Suppose . Then is a primitive th root of unity (or if ). If , then
If and is odd, then we essentially have a Gauss sum
If and is even, noting that is a th root of unity,
If then the sum will be zero. For even, this follows from the previous case. If is odd,
Now, consider the Dirichlet series
Let us combine all these facts to construct the th factor of the Dirichlet series in question, for dividing . Assume first that with even,
because for even , , and for odd , . Similarly, for odd,
Putting this together, we get that