If the counting function $N(x)$ of integers of a Beurling generalized number system satisfies both $\int _1^\infty x^{-2}|N(x)-Ax|\,dx\infty $ and $x^{-1}(\log x) (N(x)-Ax)=O(1)$, then the counting function $\pi (x)$ of the primes of this system is known to satisfy the Chebyshev bound $\pi (x)\ll x/\log x$. Let $f(x)$ increase to infinity arbitrarily slowly. We give a construction showing that $\int _1^\infty |N(x)-Ax|x^{-2}\,dx\infty $ and $x^{-1}(\log x) (N(x)-Ax)=O(f(x))$ do not imply the Chebyshev bound.
Keywords:
counting function integers beurling generalized number system satisfies int infty ax infty log ax counting function primes system known satisfy chebyshev bound log increase infinity arbitrarily slowly construction showing int infty ax infty log ax imply chebyshev bound
Affiliations des auteurs :
Harold G. Diamond
1
;
Wen-Bin Zhang
2
1
Department of Mathematics University of Illinois Urbana, IL 61801, U.S.A.
2
920 West Lawrence Ave. #1112 Chicago, IL 60640, U.S.A.
@article{10_4064_aa160_3_3,
author = {Harold G. Diamond and Wen-Bin Zhang},
title = {Optimality of {Chebyshev} bounds
for {Beurling} generalized numbers},
journal = {Acta Arithmetica},
pages = {259--275},
year = {2013},
volume = {160},
number = {3},
doi = {10.4064/aa160-3-3},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/aa160-3-3/}
}
TY - JOUR
AU - Harold G. Diamond
AU - Wen-Bin Zhang
TI - Optimality of Chebyshev bounds
for Beurling generalized numbers
JO - Acta Arithmetica
PY - 2013
SP - 259
EP - 275
VL - 160
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.4064/aa160-3-3/
DO - 10.4064/aa160-3-3
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ID - 10_4064_aa160_3_3
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%A Harold G. Diamond
%A Wen-Bin Zhang
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for Beurling generalized numbers
%J Acta Arithmetica
%D 2013
%P 259-275
%V 160
%N 3
%U http://geodesic.mathdoc.fr/articles/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 160 (2013) no. 3, pp. 259-275. doi: 10.4064/aa160-3-3
