Theory and Examples of
Generalised Prime Systems
By
Faez Ali AL-Maamori
Thesis submitted for the degree of
Doctor of Philosophy
March 2013
Abstract
A generalised prime system P is a sequence of positive reals p1, p2,p3,… satisfying
1 < p1 ≤ p2 ≤ … ≤ pn ≤ …and for which pn → ∞ as n → ∞. The {pn}
called generalised primes (or Beurling primes) with the products p1a1. P2a2…pkak
(where k ∈ N and a1,a1, … ak ∈ N ∪ {0} forming the generalised integers (or
Beurling integers).
In this thesis we study the generalised (or Beurling) prime systems and we
examine the behaviour of the generalised prime and integer counting functions
𝜋𝜌(𝑥) and 𝑁P(x) and their relation to each other, including the Beurling zeta
function 𝜁𝜌(s).
Speci cally, we study a problem discussed by Diamond (see [7]) which is to
determine the best possible in NP(x) = x + O(xe�c(log x) for some _ > 0;
given that _P(x) = li(x) + O(xe�(log x)_); _ 2 (0; 1): We obtain the result that
_ _ _:
We study the connection between the asymptotic behaviour (as x ! 1) of
the g-integer counting function NP(x) (or rather of NP(x) � ax ) and the size of
Beurling zeta function _P(_ +it) with _ near 1 (as t ! 1). We show in the _rst
section how assumptions on the growth of _P(s) imply estimates on the error term
of NP(x), while in the second half we _nd the region where _P(_ + it) = O(tc);
for some c > 0, if we assume that we have a bound for the error term of NP(x).
Finally we apply these results to _nd O and results for a speci_c example.
); for some > 0;
given that P(x) = li(x) + O(xe�(log x)
);
2 (0; 1): We obtain the result that