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It is shown that if the counting function $N$ of a generalized number system satisfies the $L^{1}$-condition $$ \\int_{1}^{\\infty}|\\frac{N(x)-ax}{x}|\\frac{\\mathrm{d}x}{x}<\\infty $$ and $N(x)=ax+o(x/\\log x),$ for some $a>0$, then $$ 0<\\liminf_{x\\to\\infty}\\frac{\\psi(x)}{x}\\ \\ \\ {and}\\ \\ \\ \\limsup_{x\\to\\infty}\\frac{\\psi(x)}{x}<\\infty $$ hold. We give an analytic proof of this result. It is based on Wiener division theorem. Our result extends those of Diamond (Proc. Amer. Math. 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