Optimality of Chebyshev bounds for Beurling generalized numbers

• Autorzy: Harold G. Diamond , Wen-Bin Zhang
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 11N80: Generalized primes and integers

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2013
• Tom: 160
• Numer: 3
• Strony: 259-275

Daty

wydano

2013

Twórcy

autor

Harold G. Diamond

• Department of Mathematics, University of Illinois, Urbana, IL 61801, U.S.A.

autor

Wen-Bin Zhang

• 920 West Lawrence Ave. #1112, Chicago, IL 60640, U.S.A.

Identyfikatory

DOI

10.4064/aa160-3-3