If the counting function N(x) of integers of a Beurling generalized number system satisfies both $∫_1^∞ x^{-2}|N(x)-Ax| dx < ∞ $ and $x^{-1}(log x)(N(x)-Ax) = O(1)$, then the counting function π(x) of the primes of this system is known to satisfy the Chebyshev bound π(x) ≪ x/logx. Let f(x) increase to infinity arbitrarily slowly. We give a construction showing that $∫_1^∞ |N(x)-Ax|x^{-2} dx < ∞$ and $x^{-1}(log x)(N(x) - Ax) = O(f(x))$ do not imply the Chebyshev bound.
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The first author conjectured that Chebyshev-type prime bounds hold for Beurling generalized numbers provided that the counting function N(x) of the generalized integers satisfies the L¹ condition $∫_1^∞ |N(x) - Ax| dx/x^2 < ∞$ for some positive constant A. This conjecture was shown false by an example of Kahane. Here we establish the Chebyshev bounds using the L¹ hypothesis and a second integral condition.
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