# A geometric approach to the odd perfect number problem?

Let $$e_d$$ be the $$d$$-th standard-basis vector in the Hilbert space $$H=l_2(\mathbb{N})$$. Let $$h(n) = J_2(n)$$ be the second Jordan totient function. Define:

$$\phi(n) = \frac{1}{n} \sum_{d|n}\sqrt{h(d)} e_d$$.

Then we have:

$$\left < \phi(a),\phi(b) \right > = \frac{\gcd(a,b)^2}{ab}=:k(a,b)$$

The vectors $$\phi(a_i)$$ are linearly independent for each finite set $$a_1,\cdots,a_n$$ of natural numbers, since

$$\det(G_n) = \prod_{i=1}^n \frac{h(a_i)}{a_i^2}$$ is not zero, where $$G_n$$ denotes the Gram matrix.

Define:

$$\hat{\phi}(n) := \sum_{d|n} \phi(d) = \frac{1}{n} \sum_{d|n} \sigma(\frac{n}{d})\sqrt{h(d)} e_d$$

Then we have:

$$n$$ is an odd perfect numbers, if and only if:

$$\left < \hat{\phi}(n),\phi(2) \right > = 1$$

By the triangle inequality we have:

$$|\hat{\phi}(n)| \le \tau(n)$$

where $$\tau$$ counts the number of divisors of $$n$$.

Geometric intuition: Since the vectors $$\phi(d), d|n$$ are almost orthogonal and have norm $$1$$, we should have by Pythagoras:

$$|\hat{\phi}(n)|^2 \approx \sum_{d|n} |\phi(d)|^2 = \tau(n)$$

A more concrete claim, which I have not been able to prove yet is: $$|\hat{\phi}(n)|^2 \ge \tau(n)$$ for all $$n$$?

Let $$\alpha$$ be the angle between $$\phi(2)$$ and $$\hat{\phi}(n)$$, where $$n$$ is an OPN. Then, by Jordans inequality for the $$\sin$$-e we get after some algebraic manipulation (and using the last claim), the following upper and lower bound for $$\tau(n)$$ for the OPN $$n$$:

$$\frac{1}{\sqrt{1-\frac{4\alpha^2}{\pi^2}}} \le \tau(n) \le \frac{1}{1-\alpha^2}$$

However it seems that numerical experiments suggest, that the last inequality can hold only for $$n=1$$ or $$n=$$ a prime, which would contradict the OPN property.

My question is, if one can prove the claim. Also asked on MO: https://mathoverflow.net/questions/373475/a-geometric-approach-to-the-odd-perfect-number-problem

Here are some notes with more details of the claims I have written above.

We have for all $$n$$:
$$|\hat{\phi}(n)|^2 = |\sum_{d|n} \phi(d)|^2 = \left < \sum_{d|n} \phi(d),\sum_{d|n} \phi(d)\right >$$ $$= \sum_{d|n} |\phi(d)|^2 + 2 \sum_{d_1 < d_2,d_1,d_2|n} \left < \phi(d_1),\phi(d_2)\right >$$ $$= \tau(n) + 2 \sum_{d_1 < d_2} \frac{\gcd(d_1,d_2)^2}{d_1 d_2}$$ $$\ge \tau(n) + 2\sum_{d_1 < d_2} \frac{1}{d_1 d_2}$$ $$\ge \tau(n) + 2\sum_{d_1 < d_2} \frac{1}{n^2}$$ $$= \tau(n) + \frac{2}{n^2} \frac{\tau(n)(\tau(n)-1)}{2}$$ $$= \tau(n) + \frac{\tau(n)(\tau(n)-1)}{n^2}$$
$$|\hat{\phi}(n)| \ge \sqrt{\tau(n)(1+\frac{\tau(n)-1}{n^2})}\ge \sqrt{\tau(n)}$$