Twin Prime Constant

The following standard mathematical constants are defined in SageMath (link):

• pi
• golden_ratio
• log2
• euler_gamma
• catalan
• khinchin
• twinprime
• mertens

I'm familiar with all the constants above except two: the twinprime constant and the khinchin constant. In this post, I'll be examining the former but first let's use SageMathCell to approximate the constants in the above list. See Figure 1.

Figure 1: permalink

Figure 2 zooms in a little closer on the output from Figure 1:

Figure 2

So what is the twin prime constant? Well, according to this source, the famous mathematical pair of Hardy and Littlewood conjectured that there are about:$$2  \prod_{p \geq 3} \frac{p(p-2)}{(p-1)^2} \int_2 ^x \frac{ \text{d}x}{(\log{x})^2}  \approx 1.320323632 \int_2 ^x \frac{ \text{d}x}{(\log{x})^2}$$twin primes less than or equal to \(x\) where the infinite product is the twin prime constant. In other words, the twin prime constant is given by$$\prod_{p \geq 3} \frac{p(p-2)}{(p-1)^2} \approx 0.66016181584686957393$$The  agreement between the calculated number of twin primes and the actual number gets better and better as \(x\) gets larger. Figure 3 shows the progression:

Figure 3

This is a complex topic and I won't go any deeper into here, as my intention was simply to explain what the constant represents and how it arises.