A Wiener-Ikehara type theorem and its application to Chebyshev bounds for Beurling primes

We provide a new version of the Wiener-Ikehara theorem where one deduces bounds $$0<\underset{x\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}\frac{S(x)}{e^x}\le \underset{x\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}\frac{S(x)}{e^x}<\mathrm{\infty }$$ for (in particular) a non-decreasing function $S$ from a mild hypothesis on the boundary behavior of its Laplace transform on a vertical segment containing $s=1$. As an application, we establish new criteria for the validity of Chebyshev bounds for Beurling generalized prime number systems under weaker conditions than were known so far.