Prime numbers, Diophantine equations, analytic number theory, algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

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Why would the roots of the generating functions of the number of k-almost primes less than x have negative real parts?

There is a 50 point bounty on this question. Specifically, I find it appealing to count only squarefree numbers having $k$ prime factors, so I define $$\pi_k(x)=\#\{n\leq x: ...