  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: $$
\mathfrak{N}(n)=n^{\frac{\log 2}{\log\log n}(1+\frac{387}{200\log\log n})}. $$
