Advanced Search

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 $-\left({\zeta }^{\text{'}}₀\left(s\right)\right)/\left(\zeta ₀\left(s\right)\right)=\zeta \left(s\right)-1$, 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...