No abstract

We deduce an asymptotic formula with error term for the sum n1,.

We study generalised prime systems P (1 < p 1 ≤ p 2 ≤ · · · , with p j ∈ R tending to infinity) and the associated Beurling zeta function ζ P (s) =Under appropriate assumptions, we establish various analytic properties of ζ P (s), including its analytic continuation and we characterise the existence of a suitable generalised functional equation. In particular, we examine the relationship between a counterpart of the Prime Number Theorem (with error term) and the properties of the analytic continuation of ζ P (s). Further we study 'well-behaved' g-prime systems, namely, systems for which both the prime and integer counting function are asymptotically well-behaved. Finally, we show that there exists a natural correspondence between generalised prime systems and suitable orders on N 2

In this paper, we study generalised prime systems for which both the prime and integer counting functions are asymptotically well-behaved, in the sense that they are approximately li(x) and x, respectively (where is a positive constant), with error terms of order O(x 1 ) and O(x 2 ) for some 1 , 2 < 1. We show that it is impossible to have both 1 and 2 less than 1 2 .

We prove that N k,ℓ=1holds for arbitrary integers 1 ≤ n 1 < · · · < n N and 0 < α < 1/2 and show by an example that this bound is optimal, up to the precise value of the exponent b(α). This estimate complements recent results for 1/2 ≤ α ≤ 1 and shows that there is no "trace" of the functional equation for the Riemann zeta function in estimates for such GCD sums when 0 < α < 1/2.