# What's the connection between $\theta$ series and the number of integer solutions on a curve? Proof Verification

I just started learning about theta series and am now flexing my muscles. I hope that everything looks good. This is a request for a proof verification. I am asking for what the proper name in the literature would be for $\mu$ defined below.

For a variable $q$ with an absolute value less than 1 we can define the absolutely convergent series. $$\mu_b(q)=\sum_{k\in\mathbb{Z}}{q^{|k|^b}}$$

This is a generalization of $\theta_3(q)=\mu_2(q)$. This would be equation $(33)$ in the link above. I am not sure what $\mu_b$ is referred to as in the literature. I suspect there is a better expression for this.

Let $\vec{a}, \vec{b}$ be tuples of $m$ positive integers $(a_1,\dots, a_m), (b_1,\dots,b_m)$ respectively. And we let $\phi_{\vec{a}}(n)$ denote the number of integer solutions to $$\sum_{i=1}^m a_i |x_i|^{b_i}=n$$.

Claim

$$\sum_{n=0}^\infty \phi_\vec{a}(n)q^n=\prod_{i=1}^m{\mu_{b_i}(q^{a_i})}$$

Proof

Starting with the RHS we need to expand the product of $m$ different doubly infinite series. We can think of this product as an $m$-dimensional array where we need to sum over $\mathbb{Z^m}$. We arrive at

$$\sum_{\vec{x}\in \mathbb{Z^m}}q^{\sum{a_i|x_i|^{b_i}}}$$

But then we need to traverse this summation somehow. For each $n\in \mathbb{N}$ we will collect the $\phi_\vec{a}(n)$ instances where $\sum_{i=1}^m a_i |x_i|^{b_i}=n$, so we arrive at the LHS.

Honestly, I think that's all that must be said for the proof but I will write it out like this:

$$\prod_{i=1}^m{\mu_{b_i}(q^{a_i})}= \prod_{i=1}^m{\sum_{k\in\mathbb{Z}}{q^{a_i|k|^{b_i}}}} = \sum_{\vec{x}\in \mathbb{Z^m}}q^{\sum{a_i|x_i|^{b_i}}}=\sum_{n=0}^\infty\sum_{\sum{a_i|x_i|^{b_i}}=n}{q^{\sum{a_i|x_i|^{b_i}}}}=\sum_{n=0}^\infty{ \phi_\vec{a}(n)q^{n}}$$

Equality 1) Definition of $\mu_{b}$

Equality 2) Multiplying $m$ doubly infinite series will give us an $m$ dimensional array. Also we use $q^Aq^B=q^{A+B}$

Equality 3) The plan for traversing this infinite sum is to start at the middle of the array and work outward. We need to be confident that we hit every single $\vec{x}\in \mathbb{Z^m}$ but this is really the same as asking if for every $\vec{x} \in \mathbb{Z^m}$ there is some $n\in \mathbb{N}$ such that $\sum_{i=1}^m a_i |x_i|^{b_i}=n$. But this boils down to closure of the natural numbers under $+$ and $\times$.

Equality 4) This is the definition of $\phi_\vec{a}(n)$.

$\square$

Questions.

1) Does everything look ok?

2) Is there a better name for $\mu_b$?

Comments/Consequence of the proof above.

If you ever spent your afternoon pondering why this OEIS sequence is entitled "Coefficients in expansion of $\theta_3(q) \times \theta_3(q^{15})$ in powers of $q$." even though it's much more intuitive to think of it as the number of integer solutions to $x^2+15y^2=n$ then you should know: Those are the same thing.

Motivations I wrote this first and then realized I could generalize.

• For example, the number of solutions to $x^2+|y|^3=n$ is given by CoefficientList[(1 + 2 Sum[q^((j)^3), {j, 10}] ) *(1 + 2 Sum[q^((j)^2), {j, 10}] ) , q] – Mason Jul 2 '18 at 21:10
• Some more motivations to asking this question are: cstheory.stackexchange.com/questions/40990/… – Mason Jul 27 '18 at 1:16

## 1 Answer

To address your questions and comments. First, your equating of the sum and product is standard and there are very many examples similar to it. Second, there is no standard name for $\, \mu_b \,$ that I know of. In the case that $\,b\,$ is odd, there is a complication that requires the use of $\, |k|^b \,$ instead of $\, k^b \,$ for convergence reasons. For example, if $\, b=1 \,$ then $\, \sum_{k\in \mathbb{Z}} q^k \,$ is never convergent. However, $\, \mu_1(q) = (1+q) / (1-q) \,$ if $\, |q|<1. \,$ It is usually unnatural to ask for the number of integer solutions to, for one example, $\, |x| + 2|y| + 3|z| = |n| \,$ although this is OEIS sequence A053799.

As for the OEIS sequence A260671 which you mentioned, it is just one of many similar sequences that are the coeffcients of functions expressed in terms of Elliptic amd related functions. Some of them have negative coefficients and because of this have no direct interpretation as enumerating solutions of Diophantine equations. In those cases where the coefficients are non-negative often no such interpretation is known to the author of the sequence, or else they forgot to mention it.

I do agree that if an interpretation of a sequence in terms of solutions to Diophanthine equations is known, then it should be entered in OEIS eventually. Thanks to your question, I have just entered another such interpretation for OEIS sequence A028625 which is directly related to my A260671.

• When is it that we can expect a nice neat closed form for the coefficients produced by these theta series? In the linked post (above under motivations) I ask for a closed form for the coefficients $\theta_3(q) \times \theta_3(q^a)$. I wonder if there is some criterion that can let us know when this is a reasonable expectation. – Mason Jul 18 '18 at 4:24
• Thank you for your answer and for all the work you put into OEIS. I am in full support of mathematicians being as accessible as is reasonable in their studies of what can be very esoteric materials. Diophantine equations are a great tool to do this as it is very accessible to the layperson. If I am digesting your answer properly: products of theta series are such a powerful tool that expressing the idea as merely solutions to Diophantine would be quite limiting. If the question above merits its own post I will happy to post it. – Mason Jul 18 '18 at 6:21