Series and Product Relations Made from Primes

Abstract

It is known that the sum of the reciprocal of integers, $\sum_n (1/n)$, and the sum of the reciprocal of primes, $\sum_n (1/p_n)$, both diverge. Here, we study a series made from primes that sums exactly to 1. We also show this sum is simply related to an infinite product over primes. We then generalize the form of the series (and its related product), extending this idea to other number theoretic sets such as twin primes. Evaluating a product made from twin primes gives a new mathematical constant.