An example in Beurling's theory of generalised primes

Faez Al-Maamori, and Titus Hilberdink. "An example in Beurling's theory of generalised primes." Acta Arithmetica 168.4 (2015): 383-395. <http://eudml.org/doc/278910>.

@article{FaezAl2015,
abstract = {We prove some connections between the growth of a function and its Mellin transform and apply these to study an explicit example in the theory of Beurling primes. The example has its generalised Chebyshev function given by [x]-1, and associated zeta function ζ₀(s) given via $- (ζ^\{\prime \}₀(s))/(ζ₀(s)) = ζ(s) - 1$, where ζ is Riemann’s zeta function. We study the behaviour of the corresponding Beurling integer counting function N(x), producing O- and Ω- results for the ’error’ term. These are strongly influenced by the size of ζ(s) near the line Re s=1.},
author = {Faez Al-Maamori, Titus Hilberdink},
journal = {Acta Arithmetica},
keywords = {generalised prime systems; Mellin transforms},
language = {eng},
number = {4},
pages = {383-395},
title = {An example in Beurling's theory of generalised primes},
url = {http://eudml.org/doc/278910},
volume = {168},
year = {2015},
}

TY - JOUR
AU - Faez Al-Maamori
AU - Titus Hilberdink
TI - An example in Beurling's theory of generalised primes
JO - Acta Arithmetica
PY - 2015
VL - 168
IS - 4
SP - 383
EP - 395
AB - We prove some connections between the growth of a function and its Mellin transform and apply these to study an explicit example in the theory of Beurling primes. The example has its generalised Chebyshev function given by [x]-1, and associated zeta function ζ₀(s) given via $- (ζ^{\prime }₀(s))/(ζ₀(s)) = ζ(s) - 1$, where ζ is Riemann’s zeta function. We study the behaviour of the corresponding Beurling integer counting function N(x), producing O- and Ω- results for the ’error’ term. These are strongly influenced by the size of ζ(s) near the line Re s=1.
LA - eng
KW - generalised prime systems; Mellin transforms
UR - http://eudml.org/doc/278910
ER -