Abstract
Kahane provided a proof of a conjecture of Bateman and Diamond on Beurling generalized primes. The conjecture says that the L2-condition [Formula: see text] implies the PNT, i.e. π(x) ~ x/ log x. Recently, Schlage-Puchta and Vindas showed that Beurling's condition in the form of a Cesàro mean [Formula: see text] with γ > 3/2 implies the PNT as well. Both are extensions of the Beurling PNT. We now show that the condition [Formula: see text] implies the PNT and that [Formula: see text] with k ≥ 0 implies the PNT also, where [Formula: see text]These results cover the preceding forms and extend further the Beurling PNT. As part of our argument, we extend first the theorems on Chebyshev bounds with correspondingly weaker conditions. These bounds play an important role in later proofs.
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