Distribution of Beurling generalized primes and integers

In 1938 Arne Beurling initiated the study of generalized number systems, which mimic the multiplicative arithmetical structure of the natural numbers without their additive structure. An increasing sequence of real numbers, starting with 1 and tending to infinity, is selected essentially arbitrarily - these we "appoint to be our primes". Then the integers are the elements of the commutative semigroup generated by these primes (using the usual multiplication of reals), ordered in increasing order. The system then satisfies the "fundamental theorem of arithmetic" if we consider two, differently represented real numbers, even if they happen to be equal, different.

Beurling proved fundamental theorems generalizing the PNT (Prime Number Theorem): if the number of integers up to x (counted acording to multiplicity) satisfies N(x)~x with a sufficiently good error term, then also the number of primes satisfy P(x)~x/log(x). The theory was further developed by J. Lagarias, H. Diamond, J.P Kahane, H. Montgomery, among others. While the arbitraryness of the choice of primes provides a great generality, where unusual constructions and new phenomena are possible, there is also a particular importance of the case when all primes, hence integers, are from N: many different structures can be investigated asymptotically in this framework. /E.g. the number of isomorphically different Abelian groups of order not exceeding x - in this counting problem our primes are the prime powers, and the fundamental theorem of Abelian groups furnishes the respective fundamental theorem of arithmetic./

In the Beurling theory a breakthrough was achieved by Diamond-Montgomery-Vorhauer. (This result was the topic of a Turán Memorial Lecture by H. Montgomery in 2000.) They basically showed that under reasonable assumptions one may not have error terms in the PNT any better than what the XIXth century result of de la Vallée-Poussin furnished for classical primes. In particular, the generalization of the Riemann Hypothesis may well fail.

In the lecture we will survey topics in part generalizing classical questions of Landau, Littlewood and Ingham. What are the pairs of exponents a,b, such that N(x)-x=O(x^a) and P(x)=x/log(x)+O(x^b) and a and b are "about best"? Can RH still fail if we have a<1? How many "exceptional zeroes" can occur, if any? If there is any exceptional zero, then what oscillation is "caused by this zero" in the error term for P(x)? Can we prescribe zeroes and poles for the zeta function at will /and still can have a Beurling system with these zeroes and poles present for zeta/? What follows, and what does not, for the behavior of P(x)-x/log(x), if we assume an error term for N(x)?