Prove $\dfrac{n^2}{\ln n^2-1} - \dfrac{\dfrac{n^2}{2}}{\ln \dfrac{n^2}{2}-1.1} > n$ for $n \ge 347$

Question

Prove $\dfrac{n^2}{\ln n^2-1} - \dfrac{\dfrac{n^2}{2}}{\ln \dfrac{n^2}{2}-1.1} > n$ for $n \ge 347$. It seems to be true for all $n \ge 11$, but I only need it to be true for $n \ge 347$. I tried to manipulate the equation in all sorts of ways but I couldn't get it to work. These are the difference of prime bounds if that helps. $\dfrac{x}{\ln x-1} < \pi(x)$ for $x \ge 5393$ and $\pi(x) < \dfrac{x}{\ln x-1.1}$ for $x \ge 60184$ where $\pi(x)$ is counts the number of primes less than or equal to $x$.

Answer 1

$$\frac{n^2}{\ln n^2-1}-\frac{\frac{n^2}{2}}{\ln\frac{n^2}{2}-1.1} \gt n \; \iff \; \frac{1}{\ln n^2-1}-\frac{1}{2(\ln\frac{n^2}{2}-1.1)} \gt \frac{1}{n} \tag{1}\label{eq1A}$$

For $n \ge 347$, we have $\ln(n) \gt 2(\ln(2) + 1.1) \; \to \; \ln(n) - \ln(2) - 1.1 \gt \frac{1}{3}\left(2\ln(n) - \ln(2) - 1.1\right)$. Using this below, we get

$$\begin{equation}\begin{aligned} \frac{1}{\ln n^2-1}-\frac{1}{2(\ln\frac{n^2}{2}-1.1)} & = \frac{1}{2\ln(n)-1}-\frac{1}{2(2\ln(n) - \ln(2)-1.1)} \\ & = \frac{2(2\ln(n) - \ln(2)-1.1) - (2\ln(n)-1)}{(2\ln(n)-1)2(2\ln(n) - \ln(2)-1.1)} \\ & = \frac{2\ln(n) - 2\ln(2)- 1.2}{(2\ln(n)-1)2(2\ln(n) - \ln(2)-1.1)} \\ & \gt \frac{2\ln(n) - 2\ln(2)- 2(1.1)}{(2\ln(n)-1)2(2\ln(n) - \ln(2)-1.1)} \\ & = \frac{\ln(n) - \ln(2)- 1.1}{(2\ln(n)-1)(2\ln(n) - \ln(2)-1.1)} \\ & \gt \frac{2\ln(n) - \ln(2)- 1.1}{3(2\ln(n)-1)(2\ln(n) - \ln(2)-1.1)} \\ & = \frac{1}{3(2\ln(n)-1)} \end{aligned}\end{equation}\tag{2}\label{eq2A}$$

Note that

$$\frac{1}{3(2\ln(n)-1)} \gt \frac{1}{n} \iff n \gt 3(2\ln(n)-1) \iff n - 6\ln(n) + 3 \gt 0 \tag{3}\label{eq3A}$$

Thus, to prove \eqref{eq1A}, it's sufficient to prove the RHS above. Define

$$f(n) = n - 6\ln(n) + 3 \tag{4}\label{eq4A}$$

We have $f(347) \approx 314.9$, and $f'(n) = 1 - \frac{6}{n} \gt 0$, so $f(n) \gt 0$ for all $n \ge 347$.

Answer 2

I tried to manipulate the equation in all sorts of ways but I couldn't get it to work

In fact, there is an exact solution to the equation $$\dfrac{n^2}{\log( n^2)-1} - \dfrac{\dfrac{n^2}{2}}{\log \left(\frac{n^2}{2}\right)-a} = n$$ which can rewrite $$n=4\frac{\left(\log (n)-\frac{1}{2}\right) (\log (n)-\alpha )}{\log (n)-\beta }$$ where

$$\alpha=\frac{1}{2} (a+\log (2)) \qquad \beta=\frac{1}{2} (2 a+2 \log (2)-1)$$

Let $n=e^x$ and write

$$e^{-x}=\frac 14 \frac{x-\beta }{\left(x-\frac{1}{2}\right) (x-\alpha )}$$ which shows an exact explicit solution in terms of the generalized Lambert function (have a look at equation $(4)$).

Since we just look for an estimate, using series $$e^x=4 x+(2 a-4+2\log (2))+O\left(\frac{1}{x}\right) $$ $$x= -\left(\frac{a}{2}+\frac{\log (2)}{2}-1\right)-W_{-1}\left(-\frac{e^{1-\frac{a}{2}}}{4 \sqrt{2}}\right)$$ giving, for $a=1.1$, $x_0=2.05486$ corresponding to $n_0=7.80572$

Newton iterates are $$\left( \begin{array}{cc} k & x_k & n_k\\ 0 & 2.05486 & 7.80572 \\ 1 & 2.33134 & 10.2917 \\ 2 & 2.30879 & 10.0623 \\ 3 & 2.30852 & 10.0595 \\ \end{array} \right)$$