Approximation of a parameter.

Question

Does $m$ have a real solution if ${t^{1+1/t}}N^{1/2m}\approx\frac {m^{1/t}}{(\log N)^{1/t}}\Big(\frac{e\log N}{\log\log N}\Big)$ where $1<t<\log m$ holds?

And if so is there a good bound for $m$?

Answer 1

Lettuce change the $\approx$ to a $=$ and rename some things. Also, lettuce consider any value of $t$, as a more general problem.

$$A\times N^{1/(2m)}=B\times m^{1/t}$$

where $A=t^{1+1/t}$ and $B=(\log N)^{-1/t}\left(\frac{e\log N}{\log\log N}\right)$.

$$\frac BA=\frac{N^{1/(2m)}}{m^{1/t}}=x^{1/t}e^{(x/2)\log(N)}\tag{$x=m^{-1}$}$$

$$\left(\frac BA\right)^t=xe^{Dx}\tag{$D=t\log(N)/2$}$$

$$\begin{align}D\left(\frac BA\right)^t & =Dxe^{Dx}\\ W\left(D\left(\frac BA\right)^t\right) & =Dx \\\end{align}$$

$$x = \frac{W\left(D\left(\frac BA\right)^t\right)}D $$

$$ m = \frac D{W\left(D\left(\frac BA\right)^t\right)}$$

$$m=\frac{t\log(N)}{2W\left(\frac{e^tt(\log N)^{t-1}}{2t^{t+1}(\log\log N)^t}\right)}$$

where $W(\dots)$ is the Lambert W function. It is continuously real if $D\left(\frac BA\right)^t>-e^{-1}$ and $N>e$ and we avoid any division by $0$.