Suppose we have a kernel $k(a,b)$ defined over the natural numbers.

Then by the Moore–Aronszajn theorem, we can embedd the natural number $a$ in some Hilbert space $\mathbb{H}$, which we write $\psi(a)$.

In the examples I am looking at we have the following:

There are some vectors $(v_a)_{a \in A}$, $v_a \in \mathbb{H}$ such that:

  1. For all $i,j \in A: \left < v_i, v_j \right > >0$

  2. For all $\epsilon > 0 \exists i,j \in A: \left < v_i,v_j \right > < \epsilon$

  3. $\forall a,b \in A: \left < v_a , v_a \right > = \left <v_b,v_b \right >$

where $A$ is some index set.

In the examples we have $v_a = \psi(a)$. It is not difficult to show, that if the vectors $(v_a)_{a \in A}$ satisfy 1) and 2) then the index set can not be finite.

I will list the examples I have in mind, some of which are not (yet?) proven to be kernels, but conjectured to be:

  1. $ k(a,b) = \dfrac{\gcd(a,b)}{a+b}$

  2. $ k(a,b) = \dfrac{1}{\mathrm{rad}\left(\frac{ab(a+b)}{\gcd(a,b)^3}\right)}$

  3. $ k(a,b) = \dfrac{1}{\sigma\left(\frac{ab}{\gcd(a,b)^2}\right)}$

  4. $ k(a,b) = \dfrac{1}{L\left(\frac{ab}{\gcd(a,b)^2}\right)}$

where $\text{rad}=$ product of prime divisors, $\sigma=$ sum of divisors, $L(n)=H_n + \exp(H_n)\log(H_n)$, $H_n=$ n-th harmonic number.

For 2) and 3) one can use that there are infinitely many primes to get the inquality $\left < v_a ,v_b \right > < \epsilon$.

For 4) one can use that the function $L$ is strictly monotonically increasing in $n$ to get $\left < v_a ,v_b \right > < \epsilon$.

Furthermore, if we insist, that $A = \mathbb{N}$ then the vectors satisfy a further, fourth property

$$4.\; \forall k,a,b \in \mathbb{N}: \left < v_{ka},v_{kb} \right > = \left < v_{a},v_{b} \right >$$

My question is, if there is anything known about such set of vectors, in this context or related context.

Thanks for your help.


Notice also the similarity between the abc-conjecture and the Riemann hypothesis as formulated by Lagarias:

If in the examples 1)-4) all are kernels, then the abc-conjecture and the Lagarias inequality might be written as "kernel-inequlaties":

$$k(a,b) \le K(a,b)$$ or $$k(a,b) < K(a,b)$$

which I find very interesting.

Related question: The abc-conjecture as an inequality for inner-products?


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