Approximation of the Multiplication Table Function

In this paper, considering the concept of Universal Multiplication Table, we show that for every $n\geq 2$, the inequality: M(n)=#\{ij|1\leq i,j\leq n\}\geq\frac{n^2}{\mathfrak{N}(n^2)}, holds true with: $$\mathfrak{N}(n)=n^{\frac{\log 2}{\log\log n}(1+\frac{387}{200\log\log n})}.$$