A discrete approach to the asymptotic enumeration of sets of primes and MacMahon's partition statistics

Abstract

Using discrete tools such as weighted and Eratosthenian counting via the inclusion-exclusion principle decorated with the M & ouml;bius function, we demonstrate how a variety of sets of primes can be shown to be quantitatively infinite. This is achieved by assuming only a one sided hypothesis in the form of an asymptotic lower bound on the number of integers which can be written as products of numbers belonging to a given set of primes. It turns out that this hypothesis goes beyond the scope of all well known trends in prime number theory. In particular, if this set of integers, whose elements are products of the given primes, contains a positive proportion of all integers, then it is shown that the number of given primes that are at most x exceeds a positive power of x. Our proofs are independent and make no use of cornerstone historical developments in prime number theory due to Chebyshev, Mertens, and we also completely evade analytic treatments like in the prime number theorems. We then apply our findings to the vanishing frequency of certain strange polynomial combinations of q-series coefficients arising from MacMahon's partition statistics. At the center of our insight and inspiration, we celebrate and develop a pretty method of Erd & odblac;s classified as one of the proofs from the Book leading to the infinitude of primes. We further introduce alternative approaches to counting primes when the Eratosthenian model is no longer sufficient.