An example in Beurling's theory of generalised primes

Faez Al-Maamori ¹   ; Titus Hilberdink ²

¹ Department of Mathematics University of Babylon Babylon, Iraq
² Department of Mathematics University of Reading Whiteknights, PO Box 220 Reading RG6 6AX, UK

Acta Arithmetica, Tome 168 (2015) no. 4, pp. 383-395

Résumé

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 $\zeta _0(s)$ given via \[ -\frac {\zeta ^{\prime }_0(s)}{\zeta _0(s)} = \zeta (s)-1,\] where $\zeta $ is Riemann's zeta function. We study the behaviour of the corresponding Beurling integer counting function $N(x)$, producing $O$- and $\varOmega $- results for the `error' term. These are strongly influenced by the size of $\zeta (s)$ near the line $\mathop {\rm Re} s=1$.

Détail

DOI : 10.4064/aa168-4-4

Keywords: prove connections between growth function its mellin transform apply these study explicit example theory beurling primes example has its generalised chebyshev function given associated zeta function zeta given via frac zeta prime zeta zeta where zeta riemanns zeta function study behaviour corresponding beurling integer counting function producing o varomega results error term these strongly influenced size zeta near line mathop