# Optimality of Chebyshev bounds for Beurling generalized numbers

Acta Arithmetica (2013)

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

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## Abstract

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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 ${x}^{-1}\left(logx\right)\left(N\left(x\right)-Ax\right)=O\left(1\right)$, 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 ${x}^{-1}\left(logx\right)\left(N\left(x\right)-Ax\right)=O\left(f\left(x\right)\right)$ do not imply the Chebyshev bound.

## How to cite

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Harold 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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