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Let $h$ be a complex valued multiplicative function. For any $N\in \mathbb{N}$, we compute the value of the determinant $D_N:={det}_{i|N,j|N}\left(\frac{h\left(\left(i,j\right)\right)}{ij}\right)$ where $\left(i,j\right)$ denotes the greatest common divisor of $i$ and $j$, which appear in increasing order in rows and columns. Precisely we prove that $$D_N=\prod _{p{}^l\parallel N}{\left(\frac{1}{p^{l\left(l+1\right)}}\stackrel{l}{\prod _{i=1}}\left(h\left(p^i\right)-h\left(p^{i-1}\right)\right)\right)}^{\tau \left(N/p^l\right)}.$$ This means that $D_N^{1/\tau \left(N\right)}$ is a multiplicative function of $N$. The algebraic apparatus associated with this result allows us to prove the following two results. The first one is the characterization of real multiplicative functions $f\left(n\right)$, with $0\le f\left(p\right)<1$, as minimal values of certain...

We express $\mathrm{\Delta }\left(x,N\right)$, as defined in the title, for $x=\frac{a}{q}$ and $q$ prime in terms of values of characters modulo $q$. Using this, we show that the universal lower bound for $\mathrm{\Delta }\left(N\right)={sup}_{x\in \mathbb{R}}\left|\mathrm{\Delta }\left(x,N\right)\right|$ can, in general, be substantially improved when $N$ is composed of primes lying in a fixed residue class modulo $q$. We also prove a corresponding improvement when $N$ is the product of the first s primes for infinitely many natural numbers $s$.