Several posts back I presented some graphs showing the number of primes and k-tuples in comparison with the numbers expected from formulae I had presented for the expected density of primes and k-tuples within primorials. The post was here:
There is a troubling systematic error in the prediction, which over-predicts the expected numbers of primes and k-tuples by some small amount (up to about 10 % toward the top of the graph for expected and actual primes, for example). I’ve been doing some further reading and found an older article at Dr. Tao’s blog about gaps in the primes that appears to shed some light on this issue. Here is the blog post:
In that post Dr. Tao refers to Merten’s theorem, which can be explored in more detail here:
I include a key quotation from Dr. Tao’s article as follows:
We then invoke Mertens’ theorem, which provides the asymptotic [formula – see Tao’s blog]. (4)
But this is off by a factor of [formula – see Tao’s blog] from what the prime number theorem says the true probability of being prime is, which is [formula – see Tao’s blog]. This discrepancy reflects the difficulty in cutting off the product in primes (4) at the right place (for instance, the sieve of Eratosthenes suggests that one might want to cut off at [formula – see Tao’s blog] instead) and I might discuss this topic further in a future blog post.
I’m intrigued by the comment about the difficulty in cutting off primes at the right place. I suspect all this ground has been covered before, long ago (Merten’s work is from the late 1800s). However, I wonder if Merten’s theorems can be exploited to make something of the evident fact that the k-tuples follow the expected formulae with similar errors as the primes, and hence must have similar properties as the primes (including infinitude).
I need to read up on Doc Merten’s work. 🙂