An example in Beurling's theory of generalised primes
Volume 168 / 2015
Acta Arithmetica 168 (2015), 383-395
MSC: Primary 11N80.
DOI: 10.4064/aa168-4-4
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 $\zeta _0(s)$ 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 $\mathop {\rm Re} s=1$.
