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An example in Beurling's theory of generalised primes

Volume 168 / 2015

Faez Al-Maamori, Titus Hilberdink 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 $\mathop {\rm Re} s=1$.

Authors

  • Faez Al-MaamoriDepartment of Mathematics
    University of Babylon
    Babylon, Iraq
    e-mail
  • Titus HilberdinkDepartment of Mathematics
    University of Reading
    Whiteknights, PO Box 220
    Reading RG6 6AX, UK
    e-mail

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