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Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

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Arguments for the second Hardy–Littlewood conjecture being false?

These are the heuristics. A. The Hardy - Littlewood conjecture for the numbers $\pi_k(x) =c_kx \log^{-k} x\left (1+o(1)\right )$ of $k$-tuples of primes $n+a_1,n+a_2,\ldots,n+a_k$ as $n \to \infty$. …
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