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# Wydawnictwa / Czasopisma IMPAN / Acta Arithmetica / Wszystkie zeszyty

## An example in Beurling's theory of generalised primes

### Tom 168 / 2015

Acta Arithmetica 168 (2015), 383-395 MSC: Primary 11N80. DOI: 10.4064/aa168-4-4

#### Streszczenie

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 $\mathop {\rm Re} s=1$.

#### Autorzy

• Faez Al-MaamoriDepartment of Mathematics
University of Babylon
Babylon, Iraq
e-mail
• Titus HilberdinkDepartment of Mathematics