Does this inequality hold true, in general?

Question

Let

$$N = \prod_{i=1}^{\omega(N)}{{p_i}^{\alpha_i}}$$

be the prime factorization of the positive integer $N$.

Does the following inequality hold true in general?

$$\prod_{i=1}^{\omega(N)}{\frac{p_i}{p_i - 1}} < \frac{2\prod_{i=1}^{\omega(N)}{p_i}}{1 + \prod_{i=1}^{\omega(N)}{p_i}}$$

If not, for what particular values of $N$ does this inequality hold?

Note that $\omega(N)$ denotes the number of distinct prime factors of $N$.

Answer 1

It turns out that that inequality is true about 95% of the time.

As in my previous answer, cancel the numerator of each side to get the equivalent inequality

$\prod_{i=1}^{\omega(N)}{\dfrac{1}{p_i - 1}} < \dfrac{2}{1 + \prod_{i=1}^{\omega(N)}{p_i}} $ or $1 + \prod_{i=1}^{\omega(N)}{p_i} < 2\prod_{i=1}^{\omega(N)}(p_i - 1) $.

Dividing by $\prod_{i=1}^{\omega(N)}{p_i}$, and, using the fact that $\dfrac{\phi(n)}{n} =\prod_{i=1}^{\omega(N)}\dfrac{p_i-1}{p_i} $, this becomes $1+\dfrac{1}{\phi(n)} <2\dfrac{\phi(n)}{n} $.

Since $\dfrac{1}{\phi(n)}$ is small for large $n$, the proportion of $n$ satisfying this is the same as the proportion satisfying $\frac{1}{2} <\dfrac{\phi(n)}{n} $, or $2 >\dfrac{n}{\phi(n)} $.

At this point, I did a Google search for "density of euler phi function". The second link is http://www.ams.org/journals/proc/2007-135-09/S0002-9939-07-08771-0/S0002-9939-07-08771-0.pdf. This paper, by ANDREAS WEINGARTNER, is titled "THE DISTRIBUTION FUNCTIONS OF σ(n)/n AND n/ϕ(n)". Here is its abstract:

"Let σ(n) be the sum of the positive divisors of n. We show that the natural density of the set of integers n satisfying σ(n)/n ≥ t is given by $\exp\big(−e^{t e^{−γ}(1 + O(t^{−2}))}\big)$ , where γ denotes Euler’s constant. The same result holds when σ(n)/n is replaced by n/ϕ(n), where ϕ is Euler’s totient function."

This paper has clearly done the heavy lifting. If we put $t = 2$, and use $\gamma \approx 0.5772156649$ (I show each stage in the computation for checkability), $te^{-\gamma} = 1.1229189671$, $e^{te^{-\gamma}}=3.0738134815$, $\exp(-e^{te^{-\gamma}}) =0.0462444658 $.

This is the density of $n$ for which $\dfrac{n}{\phi(n)} > 2 $. The density for which $\dfrac{n}{\phi(n)} < 2 $ is one minus this or $0.9537555342$.

Answer 2

We can cancel the numerator of each side to get the equivalent inequality

$\prod_{i=1}^{\omega(N)}{\dfrac{1}{p_i - 1}} < \dfrac{2}{1 + \prod_{i=1}^{\omega(N)}{p_i}} $ or $1 + \prod_{i=1}^{\omega(N)}{p_i} < 2\prod_{i=1}^{\omega(N)}(p_i - 1) $.

Let's look at the sum over primes $f(x) = \sum_{p < x} \ln (p-1) $. Since $\sum_{p < x} \ln p \approx x$ (from the prime number theorem),

$\begin{align} f(x) &=\sum_{p < x} \ln (p-1)\\ &=\sum_{p < x} (\ln p+\ln(1-1/p))\\ &=\sum_{p < x} \ln p+\sum_{p < x}\ln(1-1/p)\\ &\approx x-\sum_{p < x}(1/p+1/(2p^2)+...)\\ &\approx x-\ln \ln x+C\\ \end{align} $

From this, $e^{f(x)} \approx e^{x-\ln \ln x+C} = C_1 e^x/\ln x $.

If $N(x) = \prod_{p < x} p$ and $M(x) = \prod_{p < x} (p-1)$, $N(x) \approx e^x$ and $M(x) \approx C_1 e^x/\ln x$, $1+N(x) < 2M(x)$ means, for large enough $x$, $e^x < C_1 e^x/\ln x$, which is false.

So that inequality is often false.