Let $N = p^k m^2$ be an odd perfect number with special prime $p$.
Denote the abundancy index of the positive integer $x$ by $I(x)=\sigma(x)/x$, where $\sigma(x) = \sigma_1(x)$ is the classical sum of divisors of $x$. Additionally, denote the deficiency of $x$ by $D(x)=2x-\sigma(x)$.
I claim that the estimate $m < p$ holds if and only if $D(m) = 1$.
Recall the following material facts:
- $$\left(\frac{5}{3}\right)^{\ln(4/3)/\ln(13/9)} \approx 1.4912789897463723558$$
- $$k = 1 \Rightarrow I(m) \geq \left(I(m^2)\right)^{\ln(4/3)/\ln(13/9)} \geq \left(\frac{5}{3}\right)^{\ln(4/3)/\ln(13/9)}$$
- We have the (generic/sharp [?]) upper bound $$\frac{I(m^2)}{I(m)} \leq \frac{\zeta(2)}{\zeta(3)}.$$
- Reference: Comment by Erick Wong underneath an answer by Will Jagy to MSE question https://math.stackexchange.com/questions/1097803.
- If $I(m) < 2$, then $$\frac{2m}{m+D(m)} \leq I(m) < \frac{2m+D(m)}{m+D(m)},$$ where equality holds if and only if $I(m)=D(m)=1$.
- The inequality $p^k < m^2$, obtained by Dris in 09/2012 [JIS (2012)], means we can infer that the implication $m < p \Rightarrow k = 1$ holds.
First, we prove the following claims:
LEMMA 1. $I(m^2) \neq 2m/(m+1)$
Proof: Suppose to the contrary that $I(m^2) = 2m/(m+1)$. Since
$$I(m^2) = \frac{2}{I(p^k)} \leq \frac{2}{I(p)} = \frac{2p}{p+1},$$
then we have
$$I(m^2) = \frac{2m}{m+1} \leq \frac{2p}{p + 1}.$$
This then implies that $m < p$ (since $\gcd(p,m) = 1$). But then $m < p$ implies $k = 1$.
Consequently, we obtain
$$\frac{2m}{m+1} = I(m^2) = \frac{2}{I(p^k)} = \frac{2}{I(p)} = \frac{2p}{p+1},$$
which then gives $m = p$. This contradicts $\gcd(p,m)=1$.
LEMMA 2. $I(m^2) < 2m/(m+1)$ if and only if $p < m$.
Proof. By Lemma 1, we know that $I(m^2) \neq 2m/(m+1)$. It is obvious that $2m/(m+1) < I(m^2)$ implies $m < p$, since
$$I(m^2) = \frac{2}{I(p^k)} \leq \frac{2}{I(p)} = \frac{2p}{p+1}.$$
We want to show that $I(m^2) < 2m/(m+1)$ implies $p < m$. To this end, suppose that the inequality $I(m^2) < 2m/(m+1)$ is true. Assume that the estimate $m < p$ holds. This implies that $k = 1$, whereupon we obtain
$$\frac{2p}{p+1} = \frac{2}{I(p)} = \frac{2}{I(p^k)} = I(m^2) < \frac{2m}{m+1}.$$
Thus, we have $2p/(p+1) < 2m/(m+1)$, which implies that $p < m$, contradicting our earlier assumption that $m < p$.
It follows that $I(m^2) < 2m/(m+1) \iff p < m$, and we are done.
It is known that $D(m) = 1 \Rightarrow m < p$, since $m > 1$ is deficient, whence by a criterion for deficient numbers by Dris:
$$D(m) = 1 \iff \frac{2m}{m+1} = \frac{2m}{m+D(m)} < I(m) < \frac{2m+D(m)}{m+D(m)} = \frac{2m+1}{m+1}$$
(Use the $LHS$ inequality for $I(m)$.)
$$\bigg(\frac{2m}{m+1} < I(m) < I(m^2) \leq \frac{2p}{p+1}\bigg) \Rightarrow m < p$$
However, it is also known (by Dris [JIS (09/2012)]) that $m < p$ implies $k = 1$ (since $p^k < m^2$).
Consequently,
$$\left(D(m) = 1\right) \Rightarrow \left(m < p\right) \Rightarrow \left(k = 1\right).$$
We want to determine whether the implication $m < p \Rightarrow D(m) = 1$ is true.
Assume that $m < p$. Suppose to the contrary that $D(m) \neq 1$. (For brevity, in what follows we let $r$ denote the fraction $2m/(m+1)$.)
We consider three cases:
(1) $$\left(r < I(m) < I(m^2)\right) \Rightarrow \left((D(m) = 1) \land (m < p)\right) \Rightarrow m \text{ is almost perfect.}$$
(2) $$\left(I(m) < r < I(m^2)\right) \Rightarrow \left((D(m) \neq 1) \land (m < p)\right) \Rightarrow m \text{ is not almost perfect.}$$
(3) $$\left(I(m) < I(m^2) < r\right) \Rightarrow \left((D(m) \neq 1) \land (p < m)\right) \Rightarrow m \text{ is not almost perfect.}$$
Note that Case (3) is ruled out under the assumption $m < p$.
We consider the remaining problematic case:
Case (2): Under this case, $I(m) < r < I(m^2)$.
Since $m < p$ is true under Case (2), then we infer that (mainly because $k = 1$),
$$I(m^2) \leq \left(I(m)\right)^{\ln(13/9)/\ln(4/3)}$$
$$\frac{I(m^2)}{I(m)} \leq \left(I(m)\right)^{\ln(13/12)/\ln(4/3)}$$
where $\ln(13/12)/\ln(4/3)$ is approximately $0.27823321415675831$.
Consequently, we obtain
$$\frac{I(m^2)}{I(m)} \leq \left(I(m)\right)^{\ln(13/12)/\ln(4/3)} < \left(2\right)^{\ln(13/12)/\ln(4/3)}$$
where
$$\left(2\right)^{\ln(13/12)/\ln(4/3)} \approx 1.212708839706.$$
But since $k = 1$, then $p$ is the special prime implies that $p \geq 5$, so that
$$I(p^k) = I(p) = \frac{p+1}{p} = 1+\frac{1}{p} \leq \frac{6}{5}$$
$$I(m^2) = \frac{2}{I(p^k)} = \frac{2}{I(p)} = \frac{2p}{p+1} \geq \frac{5}{3}$$
Now consider the quantity $Q = \left(I(m^2) - r\right)\left(I(m) - r\right)$. This quantity is negative under Case (2).
Thus, we have
$$I(m^2)I(m) + r^2 < r\left(I(m^2) + I(m)\right).$$
Dividing both sides of the last inequality by $\left(I(m^2)\right)^2$, we get
$$\frac{I(m)}{I(m^2)} + \left(\frac{r}{I(m^2)}\right)^2 < \left(\frac{r}{I(m^2)}\right)\cdot\left(\frac{I(m)}{I(m^2)} + 1\right).$$
However $$\frac{I(m)}{I(m^2)} \geq \left(I(m)\right)^{\ln(4/3)/\ln(13/12)} \geq \Bigg(\left(I(m^2)\right)^{\ln(4/3)/\ln(13/9)}\Bigg)^{\ln(4/3)/\ln(13/12)}.$$
But recall from basic laws of exponents that
$$(A^{\alpha})^{\beta} = A^{\alpha\beta}.$$
Therefore, we obtain
$$\Bigg(\left(I(m^2)\right)^{\ln(4/3)/\ln(13/9)}\Bigg)^{\ln(4/3)/\ln(13/12)} = \left(I(m^2)\right)^{\frac{\left(\ln(4/3)\right)^2}{\ln(13/9)\cdot\ln(13/12)}}$$
where
$$\frac{\left(\ln(4/3)\right)^2}{\ln(13/9)\cdot\ln(13/12)} \approx 2.811777355162111571454751.$$
But our assumption $m < p$ implies that $k = 1$, which in turn is equivalent to $I(m^2)=2p/(p+1)$.
Furthermore, we compute
$$\left(\frac{r}{I(m^2)}\right)^2 = \left(\frac{(2m)/(m+1)}{(2p)/(p+1)}\right)^2 = \left(\frac{m(p+1)}{p(m + 1)}\right)^2.$$
Since $m < p$ implies $k = 1$, then the odd perfect number $\overline{N}$ takes the form $\overline{N} = pm^2$. This implies that $m < p < m^2$, whence we obtain
$$p^3 > pm^2 > {10}^{1500}$$
and
$$m^4 > pm^2 > {10}^{1500},$$
by utilizing a result of Ochem and Rao on a lower bound for the magnitude of an odd perfect number.
Thus, the estimates $p > {10}^{500}$ and $m > {10}^{375}$ both hold. Then notice that
$$\frac{r}{I(m^2)}=\frac{m(p+1)}{p(m + 1)}=\left(\frac{m}{m+1}\right)\cdot\left(\frac{p+1}{p}\right)$$
where $(p + 1)/p > 1$ (since $p$ is very large), while
$$\frac{m}{m+1}=\frac{(m+1)-1}{m+1}=1-\frac{1}{m+1}>1-\frac{1}{m}>1-{10}^{-375} > 0.999.$$
Consequently,
$$\frac{I(m)}{I(m^2)} + \left(\frac{r}{I(m^2)}\right)^2 > \left(\frac{5}{3}\right)^{\frac{\left(\ln(4/3)\right)^2}{\ln(13/9)\cdot\ln(13/12)}} + (0.999)^2$$
$$> 4.205 + 0.998 > 5.203$$
But we also know that
$$\left(\frac{r}{I(m^2)}\right)\cdot\left(\frac{I(m)}{I(m^2)} + 1\right)=\left(\frac{m}{m+1}\right)\cdot\left(\frac{p+1}{p}\right)\cdot\left(\frac{I(m)}{I(m^2)} + 1\right),$$
where
$$\frac{m}{m+1} < 1$$
$$\frac{p+1}{p}=1+\frac{1}{p} < 1 + {10}^{-1500} < 1.001,$$
and of course,
$$\frac{I(m)}{I(m^2)} + 1 < 2$$
since $I(m) < I(m^2)$ holds, in general.
Consequently,
$$\left(\frac{r}{I(m^2)}\right)\cdot\left(\frac{I(m)}{I(m^2)} + 1\right) < {1}\cdot\left(1 + {10}^{-1500}\right)\cdot{2} < 2\cdot(1.001) = 2.002.$$
We arrive at the contradiction
$$5.203 < \frac{I(m)}{I(m^2)} + \left(\frac{r}{I(m^2)}\right)^2 < \left(\frac{r}{I(m^2)}\right)\cdot\left(\frac{I(m)}{I(m^2)} + 1\right) < 2.002.$$
We therefore conclude that Case (2) does not hold.
Consequently, only Case (1) remains:
Case (1): $$r < I(m) < I(m^2) \Rightarrow \left((D(m) = 1) \land (m < p)\right) \Rightarrow m \text{ is almost perfect}.$$
In particular, $m$ is almost perfect if and only if $m < p$. (In other words, $m < p$ if and only if $D(m) = 1$.)
PostScript: It seems that we can remove the reliance of the proof on the assumption $k = 1$ (and therefore $m < p$), to obtain
$$I(m) \geq \left(I(m^2)\right)^{\frac{\ln(4/3)}{\ln(13/9)}}$$
which is equivalent to
$$I(m^2) \leq \left(I(m)\right)^{\frac{\ln(13/9)}{\ln(4/3)}},$$
from which we get
$$\frac{I(m^2)}{I(m)} \leq \left(I(m)\right)^{\frac{\ln(13/12)}{\ln(4/3)}}.$$
This last inequality implies that
$$1 > \frac{I(m)}{I(m^2)} \geq \left(I(m)\right)^{\frac{\ln(4/3)}{\ln(13/12)}}$$
$$> \Bigg(\left(I(m^2)\right)^{\frac{\ln(4/3)}{\ln(13/9)}}\Bigg)^{\frac{\ln(4/3)}{\ln(13/12)}}$$
$$= \left(I(m^2)\right)^{\frac{\left(\ln(4/3)\right)^2}{\ln(13/9)\cdot\ln(13/12)}}.$$
But in general, we know that $I(m^2) > 8/5$, so that
$$1 > \frac{I(m)}{I(m^2)} > \left(I(m^2)\right)^{\frac{\left(\ln(4/3)\right)^2}{\ln(13/9)\cdot\ln(13/12)}}> \left(\frac{8}{5}\right)^{\frac{\left(\ln(4/3)\right)^2}{\ln(13/9)\cdot\ln(13/12)}} > 3$$
which is a contradiction.
This seems to prove that there are, in fact, no odd perfect numbers.
We will stop here for the time being.
