# Optimality of Chebyshev bounds for Beurling generalized numbers

Harold G. Diamond; Wen-Bin Zhang

Acta Arithmetica (2013)

- Volume: 160, Issue: 3, page 259-275
- ISSN: 0065-1036

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topHarold G. Diamond, and Wen-Bin Zhang. "Optimality of Chebyshev bounds for Beurling generalized numbers." Acta Arithmetica 160.3 (2013): 259-275. <http://eudml.org/doc/279414>.

@article{HaroldG2013,

abstract = {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.},

author = {Harold G. Diamond, Wen-Bin Zhang},

journal = {Acta Arithmetica},

keywords = {Beurling generalized numbers; Chebyshev prime bounds; optimality},

language = {eng},

number = {3},

pages = {259-275},

title = {Optimality of Chebyshev bounds for Beurling generalized numbers},

url = {http://eudml.org/doc/279414},

volume = {160},

year = {2013},

}

TY - JOUR

AU - Harold G. Diamond

AU - Wen-Bin Zhang

TI - Optimality of Chebyshev bounds for Beurling generalized numbers

JO - Acta Arithmetica

PY - 2013

VL - 160

IS - 3

SP - 259

EP - 275

AB - 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.

LA - eng

KW - Beurling generalized numbers; Chebyshev prime bounds; optimality

UR - http://eudml.org/doc/279414

ER -

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