Chebyshev estimates for Beurling generalized prime numbers. I

We provide new sufficient conditions for Chebyshev estimates for Beurling generalized primes. It is shown that if the counting function N   of a generalized number system satisfies the L1L1-condition∫1∞|N(x)−axx|dxx<∞ and N(x)=ax+o(x/logx), for some a>0a>0, then0