$q$-analog of Number Theory

Question

The main motivation behind this is to see whether the 'magic' of q-analogs can be felt in number theory. Obviously for q-analogs to be applied to number theory the parametrization in $q$ must yield a generalization in some way whilst simultaneously keeping the structure in such a way that we are still studying numbers. For example: If in some equations involving q-numbers we are manipulating expressions and suddenly we get an additional '$q^2$ term', as so

$${[n]}_q + q^2= 1+q + 2q^2 ... + q^{n-1}$$

This becomes pretty meaningless in the context of number theory. It's good to wonder whether this is possible, studying a $q$-analog of numbers, as by constructing some q-analog without it arrising naturally we are essentially just doing number theory with some unnecessary parameters. One way I thought to do this was as follows.

Let ${[n]}_q$ denote the $q$-number such that $${[n]}_q = \frac{1-q^n}{1-q}$$ such that $\lim_{q \rightarrow 1}{[n]}_q = n$.

We have the following rule for multiplication

$${[nm]}_q = {[n]}_q{[m]}_{q^n}$$

Since we can write every number as a product of some primes and their powers $n = p_1^{k_1}p_2^{k_2}\ldots p_i^{k_i}$ we arrive at the following $q$-analog expression $${[n]}_q = {[p_1^{k_1}]}_q{[p_2^{k_2}]}_{q^{p_1^{k_1}}}\ldots{[p_i^{k_i}]}_{q^{p_1^{k_1}p_2^{k_2}\ldots p_{i-1}^{k_{i-1}}}} \\ = \left(\prod_{j=0}^{k_1-1}{[p_1]}_{q^{p_1^j}}\right)\left(\prod_{j=0}^{k_2-1}{[p_2]}_{q^{p_2^jp_1^{k_1}}}\right)\ldots\left(\prod_{j=0}^{k_i-1}{[p_i]}_{q^{p_i^jp_1^{k_1}p_2^{k_2}\ldots p_{i-1}^{k_{i-1}}}}\right)$$ resulting in the fact that $${[n]}_q = \prod_i\left(\prod_{j=0}^{\nu_{p_i}(n)-1}{[p_i]}_{q^{p_i^j\prod_{r \leq i}p_r^{k_r}}}\right)$$ (Writing the $p$-adic valuation $\nu_{p_i}(n)$ instead of $k$)

Now we can turn to cyclotomic polynomials

$$\Phi_n(x) = \prod_{\,\,\, 1 \leq k \leq n \\ \gcd(k,n) = 1} (x - e^{2\pi i \frac{k}{n}})$$ Note that since all $k \leq n$ (for some prime $n$) is coprime to $n$ if $n$ is prime we get $$\Phi_n(q) = {[n]}_q$$ (when $n$ is prime)

Because of this we can write any $q$-number as a product of cyclotomic polynomials. For example the obvious

$${[n]}_q = \prod_{d \mid n}\frac{\Phi_d(q)}{\Phi_1(q)}$$

Perhaps more interestingly, using the formula we derived above we can see that

$${[n]}_q = \prod_i\left(\prod_{j=0}^{\nu_{p_i}(n)-1}\Phi_{p_i}\left(q^{p_i^j\prod_{r \leq i}p_r^{k_r}}\right)\right)$$

Note that

$$n = \lim_{q \rightarrow 1}\prod_i\left(\prod_{j=0}^{\nu_{p_i}(n)-1}\Phi_{p_i}\left(q^{p_i^j\prod_{r \leq i}p_r^{k_r}}\right)\right)$$

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This form of product of cyclotomic polynomials has been studied by Borwein and Choi, who in their paper state that

Any polynomial $P(x)$ with even degree $N-1$ and coefficients $\pm$1 is cyclotomic iff $$P(x) = \pm \Phi_{p_1}(\pm x)\Phi_{p_2}(\pm x^{p_1})...\Phi_{p_r}(\pm x^{p_1p_2...p_{r-1}})$$ Borwein and Choi's conjecture of the odd degree case is immediately obvious for all $\quad {[2k+1]}_q \quad \forall k \in \mathbb{N}$ by the connection presented above

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At first glance this seems to offer insight into number-theoretic statements through the roots of these equations, and hence Galois theory

The twin-prime conjecture can be re-stated as asking whether there are infinitely many cases where $$1+q+q^2\Phi_p(q) = 0$$ for $q \in \{x: \Phi_{p+2}(x) = 0\}$. This could allow for a different angle to approach the conjecture, from the perspective of varieties in algebric geometry. One could derive such a polynomial in $q$ for primes with any size gap. Although I'm not sure if this would lead to any progress toward the conjecture, I do think it could reveal some interesting structures.

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For simple number sums like

$$\sum_{n=0}^{m}a_n$$ where $a_n \in \mathbb{Z}$ we have the following analog $$\sum_{n=0}^{m}{[a_n]}_qq^{\sum_{k < n}a_k}$$ This can also be considered in the case where $a_n \in \mathbb{Z}$. Either way we are left with a sum with 'Fibonacci'-like properties $$a_0+a_1q^{a_0}+a_2q^{a_0+a_1}+a_3q^{a_0+a_1+a_2}...$$ This can be studied from the perspective of Lucas sequences

If we consider the analog to a sum over the divisors of $n$ in terms of the '$q$-divisors' of $n$ (i.e the cyclotomic polynomials $\Phi_{d \mid n}(q)$) we get $$\left[n\sum_{d\mid n}\frac{f(d)}{d}\right]_q \rightarrow \left(\prod_{d\mid n}\frac{\Phi_d(q)}{\Phi_1(q)}\right) \sum_{d\mid n}\frac{f_q(d)}{\Phi_d(q)}$$ for example: $f\rightarrow (1-q)$ $$ = \Phi_{d_1}(q)\Phi_{d_2}(q)...\Phi_{d_{i-1}}(q)+...+\Phi_{d_2}(q)\Phi_{d_3}(q)...\Phi_{d_i}(q)$$

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Another potentially interesting application is the study of $q$-congruences on $q$-numbers. One could look at some

$${[n]}_q \equiv f(q) \quad \text{mod} \,\,\, \Phi_m(q)$$

$${[n]}_{q^k} \equiv \quad ... \quad \text{mod} \,\,\, {[m]}_q$$

This seems to already exist though. Here (1, 2, 3) are some papers that go over it. Although the concept seems to be studied with regards to hypergeometric series, instead of numbers, so this could also be very interesting

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These are the basic ideas. I would like to know if this parametrization and subsequent generalization of numbers in terms of cyclotomic polynomials is a viable analog to number-theory, or if there is another better analog. Relevant formulas, ideas, related papers, etc. are also appreciated.

$q$-analogs aren't widely used in (elementary) number theory, so this post (depending on the viability of 'number-theortic $q$-analogs') can serve as a starting point.

Answer 1

Here are a two somewhat related statements (given as an answer as it is too long for a comment):

• Concerning the multiplicative structure we have $$a\equiv r\pmod b \iff [a]_q \equiv [r]_q \pmod {[b]_q}$$ and as a consequence in particular $$[b]_q \mid [a]_q \iff b\mid a$$ and $$\gcd([a]_q,[b]_q) = [\gcd(a,b)]_q.$$
• Let $n$ be a non-negative integer. The subspace lattice of an $\mathbb{F}_q$-vector space of dimension $n$ is commonly considered to be the $q$-analog of the subset lattice of an $n$-element set. In particular, the $q$-analog of the binomial coefficient is the Gaussian binomial coefficient $$ \left[n\atop k\right]_q = \begin{cases}\frac{[n]_q [n-1]_q \cdots [n-k+1]_q}{[1]_q [2]_q \cdots [k]_q} & \text{if }k\in\{0,\ldots,n\} \\ 0 & \text{ otherwise,}\end{cases}$$ counting the number of $k$-dimensional subspaces of an $n$-dimensional $\mathbb{F}_q$-vector space. Note that $[{n \atop 1}]_q = [n]_q$. The Gaussian binomial coefficient $[{n\atop k}]_q$ is a polynomial in $\mathbb{Z}[q]$ of degree $k(n-k)$, and as expected by the notion "$q$-analog of the binomial coefficient", we have $[{n\atop k}]_1 = \binom{n}{k}$. All irreducible factors of $[{n\atop k}]_q\in\mathbb Z[q]$ are cyclotomic polynomials, and by the $q$-analog of the Theorem of Kummer the multiplicity of the cyclotomic polynomial $\Phi_d$ in the factorization of $[{n\atop k}]_q$ is the number of carries of the addition $k + (n-k)$ in base $d$.

Answer 2

Here are 2 problems related to cyclotomic polynomials that are unsolved :

Let $n,m>1$ and $$A(x) = \Phi_n(x) + \Phi_m(x)$$

1. If $n,m$ are distinct odd primes then $A(x)$ is irreducible.

2. If $A(x)$ factors then one of its factors is a cyclotomic polynomial.

Example :

$$\Phi_7(x) + \Phi_{22}(x) = \Phi_4(x) b(x)$$

$$\Phi_4(x) = x^2 +1 $$ $$ b(x) = x^8 - x^7 + 2 x^4 + 2$$