This project pursues new developments in the theory of Beurling generalized numbers by finding sharp conditions for the validity of major asymptotic formulas for the distribution of generalized primes. We will also establish new analytic tools and study the role of various Banach algebras of continuous functions in connection with generalized number systems and zeta functions. Our concrete goals are to:

• Obtain a general prime number theorem for Beurling generalized primes.
• Investigate remainders in the prime number theorem for number systems with low regularity.
• Give optimal conditions for the validity of Chebyshev estimates.
• Supply mild conditions for prime number theorem equivalences in the context of Beurling generalized primes.
• Develop a Banach algebra approach to various asymptotic problems in generalized prime number theory.