Series and Product Relations Made from Primes

TL;DR

This paper constructs a series from primes that sums exactly to 1, explores its relation to prime products, and extends the concept to other sets like twin primes, introducing a new constant.

Contribution

It introduces a novel prime-based series summing to 1 and generalizes it to other prime sets, revealing new mathematical constants.

Findings

A prime-based series summing to 1 is constructed.

The series is related to an infinite product over primes.

A new mathematical constant is introduced from twin 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.