I read somewhere (unfortunately I cannot find the paper again) that the twin prime counting function $\pi_2(x)$ satisfies $\pi_2(x) \leq C\frac{x}{\log^2x}$ for some constant $C$. How would one prove (or disprove) this? I can see how one would determine it heuristically with the prime number theorem but I want something more rigorous. If the formula is true, are there any bounds on $C$? Furthermore, is there an analogous formula for the number of primes $p$ such that $p + k$ is also prime, for a fixed positive integer $k$?
2025-01-13 02:36:15.1736735775
Bounds on twin prime counting function
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