The abc-conjecture as an inequality for inner-products?

Question

The abc-conjecture is:

For every $\epsilon > 0$ there exists $K_{\epsilon}$ such that for all natural numbers $a \neq b$ we have:

$$ \frac{a+b}{\gcd(a,b)}\,\ <\,\ K_{\epsilon}\cdot \text{rad}\left(\frac{ab(a+b)}{\gcd(a,b)^3}\right)^{1+\epsilon} $$

I have two questions after doing some experiments with SAGEMATH:

1. Is the matrix $$L_n = \left( \frac{\gcd(a,b)}{a+b}\right)_{1\le a,b \le n}$$ positive definite?

2. Is the matrix: $$ R_n = \left( \frac{1}{\text{rad}\left(\frac{ab(a+b)}{\gcd(a,b)^3}\right)} \right)_{1\le a,b \le n} $$ positive definite?

If both of the questions can be answered with yes, then we would have "mappings"

$$\psi ,\phi: \mathbb{N} \rightarrow \mathbb{R}^n$$

and the abc-conjecture might be stated as an inequality in the inner-product of these mappings:

$$\left< \psi(a),\psi(b) \right>^{1+\epsilon} < K_{\epsilon} \left < \phi(a), \phi(b) \right >$$

which I think would be very interesting.

Edit: I realized that it is better to ask the following question:

Is

$$R^{(\epsilon)}_n := (\frac{2^{\epsilon}}{\text{rad}(\frac{ab(a+b)}{\gcd(a,b)^3})^{1+\epsilon}})_{1\le a,b\le n}$$

positive definite for all $\epsilon \ge 0$?

If "yes", then we would have:

For all $\epsilon \ge 1$ and all $a \neq b$ the following are equivalent:

$$1) d_R^{(\epsilon)}(a,b) = \sqrt{1-\frac{2^{1+\epsilon}}{\text{rad}(\frac{ab(a+b)}{\gcd(a,b)^3})^{1+\epsilon}}}>d_L(a,b) = \sqrt{1-2\frac{\gcd(a,b)}{a+b}}$$

$$2) \left < \psi^{(\epsilon)}_R(a),\psi^{(\epsilon)}_R(b) \right > < \left < \psi_L(a),\psi_L(b) \right >$$

3. The abc conjecture for $\epsilon \ge 1$ with $K_{\epsilon} = \frac{1}{2^{\epsilon}}$

Related question Two questions around the $abc$-conjecture

Also the metrics $d_R^{(\epsilon)},d_L$ would be embedded in Euclidean space.

Yet another edit:

It seems that $$\frac{\phi(n)}{n} = \sum_{d|n} \frac{\mu(d)}{\text{rad}(d)}$$

wher $\mu, \phi$ are the Moebius function and the Euler totient function.

From this it would follow using Moebius inversion, that :

$$\frac{1}{\text{ rad}(n)} = \sum_{d|n} \frac{\mu(d)\phi(d)}{d}$$

which could (I am not sure about that) be helpful for question 2).

Edit with proof that $k(a,b)$ is a kernel: Let $$k(a,b) := \frac{1}{\frac{ab(a+b)}{\gcd(a,b)^3}} = \frac{\gcd(a,b)^3}{ab(a+b)} = \frac{\gcd(a,b)^2}{ab} \cdot \frac{\gcd(a,b)}{a+b} = k_1(a,b) \cdot k_2(a,b)$$

It is known that:

$$\int_0^1 \psi(at)\psi(bt) dt = \frac{1}{12} \frac{(a,b)^2}{ab} = \frac{1}{12} k_1(a,b).$$ Where $\psi(t) = t - \lfloor t \rfloor - \frac{1}{2}$ is the sawtooth function. Hence $k_1(a,b)$ is a kernel.

On the other hand, it is known for example by the answer of @DenisSerre, that $k_2(a,b)$ is also a kernel.

Hence the product $k(a,b) = k_1(a,b) \cdot k_2(a,b)$ is also a kernel.

Update: I found this paper online which is interesting (Set there: $X_a = \{ a/k | 1 \le k \le a \}$ then: $|X_a \cap X_b| = |X_{\gcd(a,b)}| = \gcd(a,b)$ ) and may be of use for the questions above:

https://www.researchgate.net/publication/326212690_On_the_positive_semi-definite_property_of_similarity_matrices

Setting in the paper above $A_i = \{ i/k | 1 \le k \le i \}$ we see that $|A_i \cap A_j| = |A_{\gcd(i,j)}| = \gcd(i,j)$ and $|A_i|=i$. Since in the paper it is proved that:

1. The Sorgenfrei similarity $\frac{|A_i \cap A_j|^2}{|A_i||A_j|}$ is a (positive definite $\ge0$, symmetric) kernel, we have another proof, that $\frac{\gcd(a,b)^2}{ab}$ is a kernel.

2. The Gleason similarity $\frac{2|A_i \cap A_j|}{|A_i|+|A_j|}$ is a (positive definite $\ge0$, symmetric) kernel, we have another proof, that $\frac{\gcd(a,b)}{a+b}$ is a kernel.

Using the product of these kernels, we get the new kernel $\frac{\gcd(a,b)^3}{ab(a+b)}$.

Edit, 05.03.2024: It is shown here, that the abc conjecture for $\epsilon:=1$ and $K_{\epsilon}:=1$, would follow, if it can be shown, that the symmetric function

$$K(a,b) := \frac{2(a+b)}{\gcd(a,b)\operatorname{rad}\left ( \frac{ab(a+b)}{\gcd(a,b)^3}\right )^2} $$

is a positive definite function over the natural numbers. I mention this, because it relates to the question in the following sense:

If $K$ is positive definite, the so is the function:

$$k(a,b) := \frac{\gcd(a,b)}{a+b}K(a,b)$$

as a product of two positive definite functions (where $\frac{\gcd(a,b)}{a+b}$ is positive definite as is shown by @DenisSerre ). But then, so is the function

$$\frac{1}{2}k(a,b) = \frac{1}{2} \frac{\gcd(a,b)}{a+b}K(a,b) = \frac{1}{\operatorname{rad}(\frac{ab(a+b)}{\gcd(a,b)^3})^2}$$

positive definite, which would answer for $\epsilon=1$ the question asked above positively for the matrix $R_n$.

Answer 1

The matrix $L_n$ is positive definite.

Proof. The matrix $G_n$ with entries ${\rm gcd}(a,b)$ is positive definite because of $G=D^T\Phi D$ where $\Phi={\rm diag}(\phi(1),\ldots,\phi(n))$ ($\phi$ the Euler's totient function) and $d_{ij}=1$ if $i|j$ and $0$ otherwise. Then the matrix $H_n$ with entries $\frac1{a+b}$ is positive definite because $$h_{ij}=\int_0^1 x^{i+j-1}dx$$ and the matrix with entries $x^{i+j-1}$ is positive semi-definite for $x>0$. Finally $L_n=G_n\circ H_n$ (Hadamard product) is positive definite.