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 , 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa168-4-4,
author = {Faez Al-Maamori and Titus Hilberdink},
title = {An example in Beurling's theory of generalised primes},
journal = {Acta Arithmetica},
volume = {168},
year = {2015},
pages = {383-395},
zbl = {1337.11067},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa168-4-4}
}
Faez Al-Maamori; Titus Hilberdink. An example in Beurling's theory of generalised primes. Acta Arithmetica, Tome 168 (2015) pp. 383-395. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa168-4-4/