Optimality of Chebyshev bounds for Beurling generalized numbers

Harold G. Diamond; Wen-Bin Zhang

Acta Arithmetica, Tome 161 (2013), p. 259-275 / Harvested from The Polish Digital Mathematics Library

Résumé

If the counting function N(x) of integers of a Beurling generalized number system satisfies both $\int_{1}^{\infty} x^{- 2} | N (x) - A x | d x < \infty$ and $x^{- 1} (l o g x) (N (x) - A x) = O (1)$ , 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 $\int_{1}^{\infty} | N (x) - A x | x^{- 2} d x < \infty$ and $x^{- 1} (l o g x) (N (x) - A x) = O (f (x))$ do not imply the Chebyshev bound.

Publié le : 2013-01-01

Zbl 1315.11086

EUDML-ID : urn:eudml:doc:279414

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     title = {Optimality of Chebyshev bounds for Beurling generalized numbers},
     journal = {Acta Arithmetica},
     volume = {161},
     year = {2013},
     pages = {259-275},
     zbl = {1315.11086},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa160-3-3}
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Harold G. Diamond; Wen-Bin Zhang. Optimality of Chebyshev bounds for Beurling generalized numbers. Acta Arithmetica, Tome 161 (2013) pp. 259-275. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa160-3-3/