Primitive Sets

The Erdős primitive set conjecture is that the following summation:$$ \sum _{n \, \in A} \frac {1}{n\log{n}}$$where A is any primitive set (a set where no member of the set divides another member) attains its maximum at the set of primes numbers. It was proved by Jared Duker Lichtman (pictured above) in 2022. I was informed of this via a YouTube video first released in 2022. Here is a link to the academic paper by Lichtman. The constant turns out to be about 1.6366 ... and summations of the form shown above can be no larger than this. Thus we have:$$ \sum _{p \, \in P} \frac {1}{p\log{p}} \approx 1.6366$$where P is the set of primes and \(p\) is any prime number.

Let's take the summation of the elements of the set S of semiprimes. These elements form a primitive set. Let's suppose each semiprime can be represented by its prime factors \(p\) and \(q\) where \(p \leq q\). We then have:$$ \sum _{pq \, \in S} \frac {1}{pq\log {pq}} \approx 1.1448 \dots$$We can continue this process and consider the primitive set containing all numbers with three not necessarily distinct prime factors and so on. In each case, the sum converges to a constant which can be designated as \(f_k\) where \(k\) represents the number of prime factors. Thus we've seen that \(f_1 \approx 1.6366\) and \(f_2 \approx 1.1448\). In general we write:$$f_k=\sum  \frac {1}{n\log{n}}\\ \text{where } n \text{ has } k \text{ prime factors}$$The long term behaviour of \(f_k\) is shown in Figure 1 where the term "fingerprint numbers" is used to identify these types of numbers:

Figure 1: screenshot from video

While the values of \(f_k\) initially decrease and drop below 1 for \(f_3\), it can be seen that the values bottom out around \(f_5\) and \(f_6\) and then increase slowly as they approach a value of 1 asymptotically from below. This behaviour can be seen more clearly in Figure 2.

Figure 2: screenshot from video

The term "fingerprint numbers" is an informal term used in the referenced videos and is not used formally in mathematical circles. There are many different types of primitive sets but they all have the property that no member of the set divides another member.