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Let G be a finite group and r be a prime divisor of the order of G. An irreducible character of G is said to be quasi r-Steinberg if it is non-zero on every r-regular element of G. A quasi r-Steinberg character of degree 
$\displaystyle |Syl_r(G)|$ is said to be weak r-Steinberg if it vanishes on the r-singular elements of 
$G.$ In this article, we classify the quasi r-Steinberg cuspidal characters of the general linear group 
$GL(n,q).$ Then we characterize the quasi r-Steinberg characters of 
$GL(2,q)$ and 
$GL(3,q).$ Finally, we obtain a classification of the weak r-Steinberg characters of 
$GL(n,q).$
