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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 ${\int }_1^{\infty }|N\left(x\right)-Ax|dx/x^2<\infty$ 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.

If the counting function N(x) of integers of a Beurling generalized number system satisfies both ${\int }_1^{\infty }{x}^{-2}|N\left(x\right)-Ax|dx<\infty$ and $x^{-1}\left(logx\right)(N\left(x\right)-Ax)=O\left(1\right)$, 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 ${\int }_1^{\infty }|N\left(x\right)-Ax|x^{-2}dx<\infty$ and $x^{-1}\left(logx\right)(N\left(x\right)-Ax)=O\left(f\left(x\right)\right)$ do not imply the Chebyshev bound.

Mertens’ product formula asserts that $$\prod _{p\le x}\left(1-\frac{1}{p}\right)logx\to e^{-\gamma }$$ as $x\to \infty$. Calculation shows that the right side of the formula exceeds the left side for $2\le x\le {10}^8$. It was suggested by Rosser and Schoenfeld that, by analogy with Littlewood’s result on $\pi \left(x\right)-\mathrm{li}x$, this and a complementary inequality might change their sense for sufficiently large values of $x$. We show this to be the case.