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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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An upper bound for the number of prime numbers in non-linear progressions

Yes, you can get sharp upper bound, matching (up to a multiplicative constant) the asymptotic predicted by the Bateman-Horn conjecture (https://en.wikipedia.org/wiki/Bateman%E2%80%93Horn_conjecture). …
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